BuddyPress.org

I’d remove the group feed function so I could create my own group feed loop.

What you’d want to do is remove the existing one by adding this to your theme’s functions.php:
remove_action( 'wp', 'groups_action_group_feed', 3 );

Then I’d basically duplicate the groups_action_group_feed() function (located in /buddypress/bp-groups.php), rename the function, throw it in my theme’s functions.php file and change the pointer of the following file:

bp-activity/feeds/bp-activity-group-feed.php

to a custom file I would also create in my theme’s directory.

Lastly, I would open up this duplicate of bp-activity-group-feed.php and modify the activity loop so it shows updates only from the group admin.