Group's own 'home' page, i.e. activity stream - ways to change?

For one particular project I'm working on I'd like to change the Home page for a group, i.e. that which shows the group activity. There doesn't seem to be any options about this in the BuddyPress admin.

Ideally I'd like it to be a group admin editable page. I'd like to retain the activity stream, but perhaps in a new tab within the group, so I guess the basic spec is relabel the activity streams tab (would need a diff URL too, I assume) and allow a different page to be nominated as the default page. A particular 'static' page from the group's blog (or perhaps a diff content type) would seem the obvious mechanism to deliver this.

I'm assuming this would need some custom coding/hacking. What would be recommendations for approaching this - plugin or theme? Would it make a difference if this was not used for every group?