BuddyPress.org

Check out this plugin…it will do mostly what you need, including creating a group specific blog. What more, it includes a theme for that blog and remains tied to the group, so you can pull in group information on that blog.

However, it only allows one blog per group. Check out the details here:

https://wordpress.org/extend/plugins/bp-groupblog/

Group specific forums are already inherently part of groups since BP1.1 came out.