Xavier's Secret School of Gifted Youngsters
One thing that bothers me about the Marvel Universe is the inconsistent way they apply the prejudice against mutants. For one thing, what is it about mutants that makes them more frightening that any other superhuman origin? Also, I thought that the whole "X-Men die in Dallas" bit was supposed to be the big moment where the public realizes that the X-Men are heroes. But even now, in the pages of the Avengers, you have people saying, "Why is there a mutant on the team?" Um, because there is already an alien, a robot, and a red hulk on the team.
But anyways, back in 1984 the whole fear-the-mutant storyline was in full swing, and it was touched on early in the Secret Wars series when the Avengers and the Fantastic Four are reluctant to team up with the X-Men, especially because Magneto was currently in their ranks. In the end, the X-Men declare they are a "third force" in the War, Magneto essentially excuses himself from the conflict entirely, and everyone goes about their business. Sadly, this tension didn't last and the X-Men are folded into the heroic team by the end, culminating in Colossus playing a pivotal role in their victory.
But this is my alternative storyline, and in my version the X-Men stayed out as a third party. Following the death of the Beyonder, the X-Men continued to operate outside of the post-War society of Denver and its hero teams. Even now, the X-Men are in hiding, recruiting the odd mutant child born in Denver and pursuing their own agenda. Professor X was never the most stable of personalities, and with Magneto constantly around, who knows what they might be planning to do?
In game terms, the X-Men (or whatever they might be calling themselves now) are an unknown factor, a shadow group whose motives should be suspect. Any mutant PC might be approached by the X-Men to join them rather than the Denver-based human community.