In previous work we showed that the only finite nonabelian
simple group acting by diffeomorphisms on a homology 3-
sphere is the alternating or dodecahedral group A_5. In the present paper, we characterize the finite nonsolvable groups that act on a homology
3-sphere preserving orientation. We find exactly the finite
nonsolvable groups that act orthogonally on the 3-sphere, plus
two extra families of groups for which we do not know at present if
they really can act on a homology 3-sphere.
