If a group mod its commutator subgroup is cyclic, then the group is abelian?

Question

Let $G$ be a group and let $G'$ denotes its commutator subgroup, that is the group generated by all elements of the form $g^{-1}h^{-1}gh$. Is the following true:

If $G/G'$ is cyclic, then G is abelian.

Recall that the claim is true for the $G$ modulo the center of $G$. See If $G/Z(G)$ is cyclic, then $G$ is abelian