BuddyPress.org

I strongly feel using this documentation you can achieve what you want.
https://codex.buddypress.org/developer-docs/group-extension-api/

And this should help you with manipulating navigation items in group , renaming them or removing them. (At midway of page)
http://themekraft.com/2011/02/customize-profile-and-group-menus-in-buddypress/

And this for custom fields (Not sure how much this will help for groups)
http://wparena.com/how-to/the-power-of-wordpress-custom-fields/

bye