  The only finite nonabelian simple group acting on a homology 3-sphere -
necessarily non-freely - is the dodecahedral group $\Bbb A_5 \cong {\rm
PSL}(2,5)$ (in analogy, the only finite perfect group acting freely on a
homology 3-sphere is the binary dodecahedral group $\Bbb A_5^* \cong {\rm
SL}(2,5)$). 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 $\Bbb A_5 \cong {\rm PSL}(2,5)$
and $\Bbb A_6 \cong {\rm PSL}(2,9)$. From this we deduce a short list of groups
which contains all finite nonsolvable groups admitting an action on a homology
4-spheres.
