Algorithms determining finite simple images of finitely presented groups

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for families of simple groups of Lie type of bounded rank, we obtain positive results. For example, given any fixed untwisted Lie type $X$ there is an algorithm that determines whether or not an arbitrary finitely presented group has infinitely many simple images isomorphic to $X(q)$ for some $q$, and if there are finitely many, the algorithm determines them.