Any fi{}nite group admits actions on closed 3-manifolds, and in particular
free actions. For actions with fi{}xed points, assumptions on the
type of the fi{}xed point sets of elements drastically reduce the
types of the possible groups. Concentrating on the basic case of fi{}nite
simple groups we show in the present paper that, if some involution
of a fi{}nite simple group G acting orientation-preservingly on a
closed orientable 3-manifold has nonempty connected fi{}xed point
set, then G is isomorphic to a projective linear group PSL(2, q),
and thus of a very restricted type. The question was motivated by
our work on the possible types of isometry groups of hyperbolic 3-manifolds
occuring as cyclic branched coverings of knots in the 3-sphere. We
characterize also fi{}nite groups which admit actions on $\mathbb{Z}_{2}$-homology
spheres, generalizing corresponding results for integer homology spheres.
