Spaces whose finite sheeted covers are homeomorphic to a fixed space

Finitely generated groups that have the property that all their finite index subgroups are free Abelian are shown to be either free Abelian themselves or to have prime cyclic first homology. This group theoretic result allows one to show that the first homology of a finite connected cell complex that has the property that all of its non-trivial finite index covers have total space homeomorphic to a given space must be either cyclic of prime order or free Abelian. Other topological corollaries include a classification of such 2-complexes and a classification of compact 3-manifolds that have a non-trivial finite sheeted cover and have the property that all their finite-sheeted covers are themselves.