We show that the only finite nonabelian simple groups which
admit a locally linear, homologically trivial action on a closed simply
connected 4-manifold M (or on a 4-manifold with trivial first homology) are
the alternating groups A_5, A_6 and the linear fractional group PSL(2,7) (we note that for homologically nontrivial actions all finite groups
occur). The situation depends strongly on the second Betti number b_2(M) of
M and has been known before if b_2(M) is different from two, so the main
new result of the paper concerns the case b_2(M)=2. We prove that the only
simple group that occurs in this case is A_5, and then deduce a short list
of finite nonsolvable groups which contains all candidates for actions of such
groups.
