If $G$ is finitely generated with $\mathcal{Z}(G)$ finite, $N \triangleleft G$ finite, then $\mathcal{Z}(G/N)$ is finite.

Question

I'm looking for a reference in the literature for the fact stated in the title, i.e.

If $G$ is a finitely generated group with $\mathcal{Z}(G)$ finite and $N$ is a finite normal subgroup of $G$, then $\mathcal{Z}(G/N)$ is finite.

Here $\mathcal{Z}(\cdot)$ denotes the center of a group.

I'm sure there must be some textbook on group theory that contains this result. For a proof, check the comment by Derek Holt below.