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I'm not sure about that last bit, Sam - I think your private/secret group can see your friends. I'm in a secret group, and while I'm friends with most of them, there a few that I'm not friends with. I just checked, and I can see their friends - not just the ones we share.
Regardless though - yes, you could always just make a separate FB account just for this purpose. You can even go into the settings and make it completely private - so that even if someone searched for you, they'd never find it. Or give yourself a different name online - an abbreviation of your real name, or something that you'd rather be known as.
I can't understand why anyone would need a "real" name. Unless they've found that people trying to get into such groups are unlikely to do so if they're faking and asked for their real names? I suppose that's entirely possible.
Kel