[![PSL remembers everything!][1]][1] 

O.T. O'Meara, A general isomorphism theory for linear groups, 1977.

In case where D is a division ring, you can recover both D and n from either PGL(n, D) or PSL(n, D) if $n \geq 2$. (Another way of saying the same is that nonabelian central simple algebras are uniquely determined by their abstract automorphism groups.)

When D is just a subring in division ring, situation becomes a bit trickier.

  [1]: https://i.sstatic.net/F6hrU.png