It's not going to post to your friends walls. If a friend of yours adds you to a group, it will appear on your wall (and so also on the news feeds of anyone who hasn't chosen to not see your activity) that "John Doe was added to XXX group" or similar. (I don't want to spam a friend by testing the functionality myself, but I'm certain that there's no way you can trigger a post on friends of friends walls)
The issue is that if you (as the LGBT youth) have your father as a friend, they can view your wall and will probably see that activity appear in their newsfeed.
In fact, facebook does have two different methods of reducing what someone can see. Either placing a contact in the "acquaintances" category and then only allowing "friends except acquaintances" to view posts on your wall, or manually adding an individual to the "restricted" list, which quietly blocks them from seeing anything you haven't explicitly made public would work.
The issue is that if you (as the LGBT youth) have your father as a friend, they can view your wall and will probably see that activity appear in their newsfeed.
In fact, facebook does have two different methods of reducing what someone can see. Either placing a contact in the "acquaintances" category and then only allowing "friends except acquaintances" to view posts on your wall, or manually adding an individual to the "restricted" list, which quietly blocks them from seeing anything you haven't explicitly made public would work.