The Glauberman correspondence and stable multiplicity-free subgroups

For an action by automorphisms of a finite group A on a group G of relatively prime order, the equivalence of the permutation actions of A on the set of complex irreducible characters of G and on the set of conjugacy classes of G was one of the first applications of the Glauberman–Isaacs correspondence. For such A and G, with A solvable and H an A-stable multiplicity-free subgroup of G, we show similarly, using the Glauberman correspondence and an idea of Navarro, the equivalence of the actions of A on the set of H-classes of G and on the set of pairs of irreducible characters χ of G and θ of H with χ over θ.