A reduction theorem for the Galois--McKay conjecture

We introduce {\mathcal {H}}-triples and a partial order relation on them, generalizing the theory of ordering character triples developed by Navarro and Späth. This generalization takes into account the action of Galois automorphisms on characters and, together with previous results of Ladisch and Turull, allows us to reduce the Galois-McKay conjecture to a question about simple groups.