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We are interested in finite groups acting orientation-preservingly on3-manifolds (arbitrary actions, ie not necessarily free actions). In particularwe consider finite groups which contain an involution with nonempty connectedfixed point set. This condition is satisfied by the isometry group of anyhyperbolic cyclic branched covering of a strongly invertible knot as well as bythe isometry group of any hyperbolic 2-fold branched covering of a knot in the3-sphere. In the paper we give a characterization of nonsolvable groups of thistype. Then we consider some possible applications to the study of cyclicbranched coverings of knots and of hyperelliptic diffeomorphisms of3-manifolds. In particular we analyze the basic case of two distinct knots withthe same cyclic branched covering.

Finite groups acting on 3-manifolds and cyclic branched coverings of knots