Groups with finitely generated integral homologies

Suppose $A$ is an abelian normal subgroup of a finitely generated group $G$ such that $G/A$ is abelian and $H_i(G, \BZ)$ is finitely generated for all $i$ . We show that $A$ is of finite (Pr\"ufer) rank. This generalises the main result of \cite{gr} that deals with the same problem for split extension metabelian groups.