In this paper we consider hyperbolic manifolds which are $p$-fold cyclic branched coverings of a link where $p$ is a prime number different from two; we prove that the Sylow $p$-subgroups of the orientation-preserving isometry groups of these 3-manifolds are either abelian of rank at most three or $p$ is equal to three and a Sylow $p$-subgroup is a $\bb{Z}_3$-extension of an abelian subgroup of rank three. As a corollary of this result we obtain that there are at most three inequivalent links which have the same hyperbolic 3-manifold as a $p$-fold cyclic branched covering and we describe the relation between two such links.