  We show that a finite group which admits a faithful, smooth,
orientation-preserving action on a homology 4-sphere, and in particular on the
4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5), by
explicitly determining the various groups which can occur (up to an
indetermination of index two in the case of solvable groups). As a consequence
we obtain also a characterization of the finite groups which are isomorphic to
subgroups of the orthogonal groups SO(5) and O(5).
