Character sums and double cosets

If G is a p-solvable finite group, P is a self-normalizing Sylow p-subgroup of G with derived subgroup P′, and Ψ is the sum of all the irreducible characters of G of degree not divisible by p, then we prove that the integer Ψ(P′zP′) is divisible by |P| for all z∈G. This answers a question of J. Alperin.