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|| = Π₏ ⏟ (₆) |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 π (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.