Thanks for this thread. As someone who has used Groups primarily to game with, I'll apply that lens to illustrate why I would prefer slightly different choices.
* The group setting should override the profile identity, so that the player can freely join nay group, and be known as themselves, but for certain groups they choose, always be seen as their alias within that group. Aliases (especially in gaming) vary for each group. This logic would seem to be more beneficial to users on any given group as well, who may prefer to be known in one group by one alias, but by a very different one elsewhere.
* agree that aliases should not transfer between groups. If you want the same alias between multiple groups either set up a new alias on each group. or else change your profile and only put aliases where you don't want it, which ever is less hassle for you personally.
* As far as mods & settings go, I would say having these options available within each group make sense to me:
WIth these options in place, groups can be run the way that suits them best. I guess you could technically drop option D from the list, and just set it to option A and never add aliases, but then the preference would be less clear on the front page. "Disabled" lets people know up front that it's not allowed.