BuddyPress.org

If you create a custom post type called groups, or whatever you have chosen for a groups slug and then create a custom post type titled the same as a group name you can use the post editor in the admin to update the home page of any group. Basically you are creating the same URL to this content.

This will remove the activity from the front page of your groups. Create a file in your themes folder called front.php.

themes folder/groups/single/front.php

Then in front.php just add a wordpress loop or any other content you wish. You can use the single.php loop or create a loop for your custom post type.