The commuting graphs of certain cyclic-by-abelian groups

Let $G$ be a finite, non-abelian group of the form $G=AN$, where $A\le G$ is abelian, and $N⊴G$ is cyclic. We prove that the commuting graph $\mathrm{\Gamma }(G)$ of $G$ is either a connected graph of diameter at most four, or the disjoint union of $\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }+1$ complete graphs. These results apply to all finite metacyclic groups, and to groups of square-free order in particular.