Character correspondences above fully ramified sections and Schur indices

Let N be a finite group of odd order and A a finite group that acts on N such that the orders of N and A are coprime. Isaacs constructed a natural correspondence between the set Irr_A(N) of irreducible complex characters invariant under the action of A, and the irreducible characters of the centralizer of A in N, Irr(C_N(A)). We show that this correspondence preserves Schur indices over the rational numbers. Moreover, suppose that the semidirect product AN is a normal subgroup of the finite group G and set U= N_G(A). Let \chi \in Irr_A(N) and \chi* \in Irr(C_N(A)) correspond. Then there is a canonical bijection between Irr(G | \chi) and Irr(U | \chi*) preserving Schur indices. We also give simplified and more conceptual proofs of (known) character correspondences above fully ramified sections.