On Certain Automorphism Groups of Finitely Generated Groups

Let $G$ be a group, and let $\operatorname{Hom}(G,N)$ be the group of all homomorphisms of $G$ into an Abelian subgroup $N$ of $G$. We give here a necessary condition for finitely generated groups to satisfy the condition that $\operatorname{Hom}(G/L,N)$ is isomorphic to $G/M$, where $L\le M$, $L$ and $M$ are normal subgroups of $G$. Consequently, we also extend some existing results on equality of two automorphism groups.