How to find G-admissible abelian regular coverings of a graph?

Given a finite connected simple graph Γ, and a subgroup G of its automorphism group, a general method for finding all finite abelian regular coverings of Γ that admit a lift of each element of G is developed. As an application, all connected arc-transitive abelian regular coverings of the Petersen graph are classified up to isomorphism.