Finite groups acting on hyperelliptic 3-manifolds

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to S3. Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on 3-manifolds and containing hyperelliptic involutions whose fixed-point set has r > 2 components. In particular we prove that a simple group containing such an involution is isomorphic to PSL(2,q) for some odd prime power q, or to one of four other small simple groups.