Isomorphisms of Cayley graphs of a free Abelian group

A group $G$ is called a $CI$-group provided that the existence of some automorphism $\sigma\in\operatorname{Aut}(G)$, such that $\sigma(A)=B$ follows from an isomorphism $\operatorname{Cay}(G,A)\cong\operatorname{Cay}(G,B)$ between Cayley graphs, where $A$ and $B$ are two systems of generators for $G$. We prove that every finitely generated abelian group is a $CI$-group.