BuddyPress.org

I’m not sure that this is an issue. If you look at the tables in the DB, you will see that there is no specific group-activity stream.

What you are seeing is the activity of each of the group’s members being reported. Since they are members of that group, their activity anywhere in your site is reported. It is, in essence, a record of what this group’s members are doing–inside and outside of the group.