  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.
