The Jacobian group of a graph is a finite abelian group through which we can study the graph in an algebraic way. When the graph is a finite abelian covering of another graph, the Jacobian group is equipped with the action of the Galois group. In this paper, we study the Fitting ideal of the Jacobian group as a module over the group ring. We also study the corresponding question for infinite coverings. Additionally, this paper includes module-theoretic approach to Iwasawa theory for graphs.