In 1942 the world was was at war with a dictator who was running experiments humanity had never heard of and probably never would. 

Our government though heard of these experiments and started it's own branch to match or beat the German's at their own game.  The branch was called GSA (Genetic Security Agency).

It's job was to learn how to unlock the human DNA and manipulate it as it desired  It had some success and mostly failure.  Till a young man came along. 

By 1974 Adam McCoy was a young 15 something scientist whose IQ knew no bounds.  He wanted to help the world, not hurt it.  But like most young men at his age, he was unusually trusting, especially of the U.S. Government.  He was asked to come and help, his job was to find cures for diseases, the thinking was if he manipulated DNA to find cures, they could take his work and turn it into anything they wanted.

It worked too.  Adam was soon working for them and making great strides in curing mostly children of incurable diseases, and occasionally adults. 

Unknown to him, they were taking his work and using it on other subjects as well.  The Government wanted their weapons, not knowing what was going to happen in a few years to those children Adam cured. 

Adam worked there for over 15 years, before he started to see these children he'd cured.  They were brought back to GSA and most were turned to what the government wanted.  It took Adam a few to realize his good work had been turned towards evil, at least in his eyes. 

As the children grew they started to exhibit abilities, mostly by the time they put puberty.  Their abilities sometimes matched those of the projects Adam was also finding out about. 

Unable to take it any more, Adam rescued as many as he could, both adults and children and took what money he'd made over the years and hid out.  It took a bit longer to find and buy Alcatraz Island, but soon his base was set up and he was rescuing more mutants from the government who strove to use them as weapons.

Along the way he found that not all the U.S. Government was behind the mutants, but the part that was willing to help Adam and his crew couldn't step in and stop the GSA and their affiliates without letting the world know of the mutants, something all were scared of.

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Remember there are only 3 ways for a mutant to come into being.  You must have been cured by Adam McCoy, sometime after 1974 (either as a child or adult.)

You were experimented on by the US Government and turned into one of the Scientific mutants...that can be before 74, but successes then were rare. 

Or you are the child of one of the two.  Again a parent can be an experiment before 74, but again its rare they'd survive for long after.