The only finite nonabelian simple group acting on a homology 3-sphere - necessarily non-freely - is the dodecahedral group A_5 isomorphic to PSL(2,5) (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group SL(2,5)). In the present paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, are the alternating or linear fractional groups groups A_5 isomorphic to PSL(2,5) and A_6 isomorphic to PSL(2,9). From this we deduce a short list of groups which contains all finite nonsolvable groups admitting an action on a homology 4-spheres.