  Let $G$ be a finitely generated group and let $C^*$ denote the group of all
central automorphisms of $G$ fixing the center of $G$ elementwise. Azhdari and
Malayeri [J. Algebra Appl., {\bf 6}(2011), 1283-1290] gave necessary and
sufficient conditions on $G$ such that $C^* \simeq \mathrm{Inn}(G)$. We prove a
technical lemma and, as a consequence, obtain a short and easy proof of this
result of Azhdari and Malayeri. Subsequently, we also obtain short proofs of
some other existing and some new related results.
