Existence and uniqueness of group structures on covering spaces over groups

Let f : X → Y be a covering map from a connected space X onto a topological group Y and let x₀ ∈ X be a point such that f(x₀) is the identity of Y: We examine if there exists a group operation on X which makes X a topological group with identity x₀ and f a homomorphism of groups. We prove that the answer is positive in two cases: if f is an overlay map over a locally compact group Y, and if Y is locally compactly connected. In this way we generalize previous results for overlay maps over compact groups and covering maps over locally path-connected groups. Furthermore, we prove that in both cases the group structure on X is unique.