Variations on a theorem by Alan Camina on conjugacy class sizes

Let G be a finite group. We extend Alan Caminaʹs theorem on conjugacy class sizes which asserts that if the conjugacy class sizes of G are exactly {1,pa,qb,paqb} for two primes p and q, then G is nilpotent. If we assume that G is solvable, we show that when the set of conjugacy class sizes of G is {1,m,n,mn} with m and n arbitrary positive integers such that (m,n)=1, then G is nilpotent and m=pa and n=qb for two primes p and q.