An infinite family of counterexamples to a question of Camina

A.R. Camina and R.D. Camina posed the following question: Suppose there are two finite groups, one nilpotent and the other non-nilpotent, and the two groups share identical sets of conjugacy class sizes; must the non-nilpotent group possess a non-trivial center? Recently, W. Zhou gave a negative answer via a subtle and elegant construction of concrete counterexamples. Nevertheless, his approach relies on the existence of Sophie Germain primes, and thus fails to yield infinitely many counterexamples unconditionally. In the present paper, we construct an infinite family of counterexamples to Camina’s question.