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

Given a finite connected simple graph $\Gamma$, and a subgroup $G$ of its automorphism group, a general method for finding all finite abelian regular coverings of $\Gamma$ 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.