Let $G$ be a finitely generated group and let $C$ be an infinite cyclic normal subgroup of $G$. My question is: Does $G$ necessarily have a finite-index subgroup $H$ such that $C \le Z(H)$? If not, then are there well-known assumptions that make it so?
I have an idea for an argument, but I'm afraid I might be misunderstanding some things about subgroup distortion, so I'm asking here in case someone knows the answer for sure.