Character correspondences in solvable groups

Let G be a finite solvable group, p be some prime, let P be a Sylow p-subgroup of G, and let N be its normalizer in G. It is known that the number of irreducible characters of G of degree prime to p equals the number of irreducible characters of N of degree prime to p. Let F be any field of characteristic zero such that, if [G:N] is even, then F contains Qp∩Q(ζ2∞), the intersection of the field of p-adic numbers with the field of rational numbers extended by all roots of unity whose order is a power to two. In this paper, we show that there exists a bijection from the set of all irreducible characters of G of degree prime to p to the set of all the irreducible characters of degree prime to p of N such that it preserves ± the degrees modulo p, all the field of values over F, and the Schur index over every field containing F.