Try adding an inner container to the Group block.

packages/block-library/src/group/deprecated.js

@@ -0,0 +1,48 @@
+/**
+ * External dependencies
+ */
+import classnames from 'classnames';
+
+/**
+ * WordPress dependencies
+ */
+import { InnerBlocks, getColorClassName } from '@wordpress/block-editor';
+
+const deprecated = [
+	// v1 of group block. Deprecated to add an inner-container div around `InnerBlocks.Content`.
+	{
+		attributes: {
+			backgroundColor: {
+				type: 'string',
+			},
+			customBackgroundColor: {
+				type: 'string',
+			},
+		},
+		supports: {
+			align: [ 'wide', 'full' ],
+			anchor: true,
+			html: false,
+		},
+		save( { attributes } ) {
+			const { backgroundColor, customBackgroundColor } = attributes;
+
+			const backgroundClass = getColorClassName( 'background-color', backgroundColor );
+			const className = classnames( backgroundClass, {
+				'has-background': backgroundColor || customBackgroundColor,
+			} );
+
+			const styles = {
+				backgroundColor: backgroundClass ? undefined : customBackgroundColor,
+			};
+
+			return (
+				<div className={ className } style={ styles }>
+					<InnerBlocks.Content />
+				</div>
+			);
+		},
+	},
+];
+
+export default deprecated;

packages/block-library/src/group/edit.js

@@ -45,9 +45,11 @@ function GroupEdit( {
 				/>
 			</InspectorControls>
 			<div className={ classes } style={ styles }>
-				<InnerBlocks
-					renderAppender={ ! hasInnerBlocks && InnerBlocks.ButtonBlockAppender }
-				/>
+				<div className="wp-block-group__inner-container">
+					<InnerBlocks
+						renderAppender={ ! hasInnerBlocks && InnerBlocks.ButtonBlockAppender }
+					/>
+				</div>
 			</div>
 		</>
 	);

packages/block-library/src/group/editor.scss

@@ -11,14 +11,14 @@
 	}
 
 	// Only applied when background is added to cancel out padding
-	> .editor-block-list__block-edit > div > .wp-block-group.has-background > .editor-inner-blocks {
+	> .editor-block-list__block-edit > div > .wp-block-group.has-background > .wp-block-group__inner-container > .editor-inner-blocks {
 		margin-top: -#{$block-padding*2 + $block-spacing};
 		margin-bottom: -#{$block-padding*2 + $block-spacing};
 	}
 
 	// Full Width Blocks
 	// specificity required to only target immediate child Blocks of a Group
-	> .editor-block-list__block-edit > div > .wp-block-group > .editor-inner-blocks > .editor-block-list__layout > .wp-block[data-align="full"] {
+	> .editor-block-list__block-edit > div > .wp-block-group > .wp-block-group__inner-container > .editor-inner-blocks > .editor-block-list__layout > .wp-block[data-align="full"] {
 		margin-left: auto;
 		margin-right: auto;
 		padding-left: $block-padding*2;
@@ -31,7 +31,7 @@
 	}
 
 	// Full Width Blocks with a background (ie: has padding)
-	> .editor-block-list__block-edit > div > .wp-block-group.has-background > .editor-inner-blocks > .editor-block-list__layout > .wp-block[data-align="full"] {
+	> .editor-block-list__block-edit > div > .wp-block-group.has-background > .wp-block-group__inner-container > .editor-inner-blocks > .editor-block-list__layout > .wp-block[data-align="full"] {
 		// note: using position `left` causes hoz scrollbars so
 		// we opt to use margin instead
 		// the 30px matches the hoz padding applied in `theme.scss`
@@ -51,7 +51,7 @@
 .wp-block[data-type="core/group"][data-align="full"] {
 
 	// First tier of InnerBlocks must act like the container of the standard canvas
-	> .editor-block-list__block-edit > div > .wp-block-group > .editor-inner-blocks {
+	> .editor-block-list__block-edit > div > .wp-block-group > .wp-block-group__inner-container > .editor-inner-blocks {
 		margin-left: auto;
 		margin-right: auto;
 		padding-left: 0;
@@ -65,7 +65,7 @@
 
 	// Full Width Blocks
 	// specificity required to only target immediate child Blocks of Group
-	> .editor-block-list__block-edit > div > .wp-block-group > .editor-inner-blocks > .editor-block-list__layout > .wp-block[data-align="full"] {
+	> .editor-block-list__block-edit > div > .wp-block-group > .wp-block-group__inner-container > .editor-inner-blocks > .editor-block-list__layout > .wp-block[data-align="full"] {
 		padding-right: 0;
 		padding-left: 0;
 		left: 0;
@@ -81,7 +81,7 @@
 
 	// Full Width Blocks with a background (ie: has padding)
 	// note: also duplicated above for all Group widths
-	> .editor-block-list__block-edit > div > .wp-block-group.has-background > .editor-inner-blocks > .editor-block-list__layout > .wp-block[data-align="full"] {
+	> .editor-block-list__block-edit > div > .wp-block-group.has-background > .wp-block-group__inner-container > .editor-inner-blocks > .editor-block-list__layout > .wp-block[data-align="full"] {
 		width: calc(100% + 60px);
 	}
 }

packages/block-library/src/group/index.js

@@ -6,6 +6,7 @@ import { __ } from '@wordpress/i18n';
 /**
  * Internal dependencies
  */
+import deprecated from './deprecated';
 import edit from './edit';
 import icon from './icon';
 import metadata from './block.json';
@@ -27,4 +28,5 @@ export const settings = {
 	},
 	edit,
 	save,
+	deprecated,
 };

packages/block-library/src/group/save.js

@@ -22,7 +22,9 @@ export default function save( { attributes } ) {
 
 	return (
 		<div className={ className } style={ styles }>
-			<InnerBlocks.Content />
+			<div className="wp-block-group__inner-container">
+				<InnerBlocks.Content />
+			</div>
 		</div>
 	);
 }

packages/e2e-tests/fixtures/blocks/core__group.html

@@ -1,10 +1,12 @@
 <!-- wp:group {"backgroundColor":"secondary","align":"full"} -->
 <div class="wp-block-group alignfull has-secondary-background-color has-background">
-	<!-- wp:paragraph -->
-	<p>This is a group block.</p>
-	<!-- /wp:paragraph -->
+	<div class="wp-block-group__inner-container">
+		<!-- wp:paragraph -->
+		<p>This is a group block.</p>
+		<!-- /wp:paragraph -->
 
-	<!-- wp:paragraph -->
-	<p>Group block content.</p>
-	<!-- /wp:paragraph --></div>
+		<!-- wp:paragraph -->
+		<p>Group block content.</p>
+		<!-- /wp:paragraph --></div>
+	</div>
 <!-- /wp:group -->

packages/e2e-tests/fixtures/blocks/core__group.json

@@ -31,6 +31,6 @@
                 "originalContent": "<p>Group block content.</p>"
             }
         ],
-        "originalContent": "<div class=\"wp-block-group alignfull has-secondary-background-color has-background\">\n\t\n\n\t</div>"
+        "originalContent": "<div class=\"wp-block-group alignfull has-secondary-background-color has-background\">\n\t<div class=\"wp-block-group__inner-container\">\n\t\t\n\n\t\t</div>\n\t</div>"
     }
 ]

packages/e2e-tests/fixtures/blocks/core__group.parsed.json

@@ -10,28 +10,28 @@
                 "blockName": "core/paragraph",
                 "attrs": {},
                 "innerBlocks": [],
-                "innerHTML": "\n\t<p>This is a group block.</p>\n\t",
+                "innerHTML": "\n\t\t<p>This is a group block.</p>\n\t\t",
                 "innerContent": [
-                    "\n\t<p>This is a group block.</p>\n\t"
+                    "\n\t\t<p>This is a group block.</p>\n\t\t"
                 ]
             },
             {
                 "blockName": "core/paragraph",
                 "attrs": {},
                 "innerBlocks": [],
-                "innerHTML": "\n\t<p>Group block content.</p>\n\t",
+                "innerHTML": "\n\t\t<p>Group block content.</p>\n\t\t",
                 "innerContent": [
-                    "\n\t<p>Group block content.</p>\n\t"
+                    "\n\t\t<p>Group block content.</p>\n\t\t"
                 ]
             }
         ],
-        "innerHTML": "\n<div class=\"wp-block-group alignfull has-secondary-background-color has-background\">\n\t\n\n\t</div>\n",
+        "innerHTML": "\n<div class=\"wp-block-group alignfull has-secondary-background-color has-background\">\n\t<div class=\"wp-block-group__inner-container\">\n\t\t\n\n\t\t</div>\n\t</div>\n",
         "innerContent": [
-            "\n<div class=\"wp-block-group alignfull has-secondary-background-color has-background\">\n\t",
+            "\n<div class=\"wp-block-group alignfull has-secondary-background-color has-background\">\n\t<div class=\"wp-block-group__inner-container\">\n\t\t",
             null,
-            "\n\n\t",
+            "\n\n\t\t",
             null,
-            "</div>\n"
+            "</div>\n\t</div>\n"
         ]
     },
     {

packages/e2e-tests/fixtures/blocks/core__group.serialized.html

@@ -1,9 +1,9 @@
 <!-- wp:group {"backgroundColor":"secondary","align":"full"} -->
-<div class="wp-block-group alignfull has-secondary-background-color has-background"><!-- wp:paragraph -->
+<div class="wp-block-group alignfull has-secondary-background-color has-background"><div class="wp-block-group__inner-container"><!-- wp:paragraph -->
 <p>This is a group block.</p>
 <!-- /wp:paragraph -->
 
 <!-- wp:paragraph -->
 <p>Group block content.</p>
-<!-- /wp:paragraph --></div>
+<!-- /wp:paragraph --></div></div>
 <!-- /wp:group -->

packages/e2e-tests/fixtures/blocks/core__group__deprecated.html

@@ -0,0 +1,7 @@
+<!-- wp:group {"backgroundColor":"lighter-blue","align":"full"} -->
+<div class="wp-block-group alignfull has-lighter-blue-background-color has-background">
+	<!-- wp:paragraph -->
+	<p>test</p>
+	<!-- /wp:paragraph -->
+</div>
+<!-- /wp:group -->

packages/e2e-tests/fixtures/blocks/core__group__deprecated.json

@@ -0,0 +1,25 @@
+[
+    {
+        "clientId": "_clientId_0",
+        "name": "core/group",
+        "isValid": true,
+        "attributes": {
+            "backgroundColor": "lighter-blue",
+            "className": "alignfull"
+        },
+        "innerBlocks": [
+            {
+                "clientId": "_clientId_0",
+                "name": "core/paragraph",
+                "isValid": true,
+                "attributes": {
+                    "content": "test",
+                    "dropCap": false
+                },
+                "innerBlocks": [],
+                "originalContent": "<p>test</p>"
+            }
+        ],
+        "originalContent": "<div class=\"wp-block-group alignfull has-lighter-blue-background-color has-background\">\n\t\n</div>"
+    }
+]

packages/e2e-tests/fixtures/blocks/core__group__deprecated.parsed.json

@@ -0,0 +1,35 @@
+[
+    {
+        "blockName": "core/group",
+        "attrs": {
+            "backgroundColor": "lighter-blue",
+            "align": "full"
+        },
+        "innerBlocks": [
+            {
+                "blockName": "core/paragraph",
+                "attrs": {},
+                "innerBlocks": [],
+                "innerHTML": "\n\t<p>test</p>\n\t",
+                "innerContent": [
+                    "\n\t<p>test</p>\n\t"
+                ]
+            }
+        ],
+        "innerHTML": "\n<div class=\"wp-block-group alignfull has-lighter-blue-background-color has-background\">\n\t\n</div>\n",
+        "innerContent": [
+            "\n<div class=\"wp-block-group alignfull has-lighter-blue-background-color has-background\">\n\t",
+            null,
+            "\n</div>\n"
+        ]
+    },
+    {
+        "blockName": null,
+        "attrs": {},
+        "innerBlocks": [],
+        "innerHTML": "\n",
+        "innerContent": [
+            "\n"
+        ]
+    }
+]

packages/e2e-tests/fixtures/blocks/core__group__deprecated.serialized.html

@@ -0,0 +1,5 @@
+<!-- wp:group {"backgroundColor":"lighter-blue","className":"alignfull"} -->
+<div class="wp-block-group has-lighter-blue-background-color has-background alignfull"><div class="wp-block-group__inner-container"><!-- wp:paragraph -->
+<p>test</p>
+<!-- /wp:paragraph --></div></div>
+<!-- /wp:group -->

packages/e2e-tests/specs/blocks/__snapshots__/group.test.js.snap

@@ -2,20 +2,20 @@
 
 exports[`Group can be created using the block inserter 1`] = `
 "<!-- wp:group -->
-<div class=\\"wp-block-group\\"></div>
+<div class=\\"wp-block-group\\"><div class=\\"wp-block-group__inner-container\\"></div></div>
 <!-- /wp:group -->"
 `;
 
 exports[`Group can be created using the slash inserter 1`] = `
 "<!-- wp:group -->
-<div class=\\"wp-block-group\\"></div>
+<div class=\\"wp-block-group\\"><div class=\\"wp-block-group__inner-container\\"></div></div>
 <!-- /wp:group -->"
 `;
 
 exports[`Group can have other blocks appended to it using the button appender 1`] = `
 "<!-- wp:group -->
-<div class=\\"wp-block-group\\"><!-- wp:paragraph -->
+<div class=\\"wp-block-group\\"><div class=\\"wp-block-group__inner-container\\"><!-- wp:paragraph -->
 <p>Group Block with a Paragraph</p>
-<!-- /wp:paragraph --></div>
+<!-- /wp:paragraph --></div></div>
 <!-- /wp:group -->"
 `;