  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.
