Nilpotency of p-complements and p-regular conjugacy class sizes

Let G be a finite p-solvable group. We prove that if the set of conjugacy class sizes of all p′-elements of G is {1,m,pa,mpa}, where m is a positive integer not divisible by p, then the p-complements of G are nilpotent and m is a prime power. This result partially extends a theorem for ordinary classes which asserts that if the set of conjugacy class sizes of a finite group G is exactly {1,m,n,mn} and (m,n)=1, then G is nilpotent.