On $A$-Groups with the Same Index Set as a Nilpotent Group

Let $G$ be a finite group and $N(G)$ be the set of conjugacy class sizes of $G$. For a prime $p$, let $|G||_p$ be the highest $p$-power dividing some element of $N(G)$. and define $|G|| = {\Pi}_{p\in {\pi}(G)}|G||_p$. $G$ is said to be an $A$-group if all its Sylow subgroups are abelian. We prove that if $G$ is an $A$-group such that $N(G)$ contains $|G||_p$ for every $p\in {\pi}(G)$ as well as $|G||$, then $G$ must be abelian. This result gives a positive answer to a question posed by Camina and Camina in 2006.