The McKay conjecture was refined in 2004 by G. Navarro taking into account the action of Galois automorphisms on characters. This refinement is known as the Galois McKay conjecture, and it is one of the last counting conjectures without a reduction theorem. In this talk I will present a possible reduction theorem for the Galois McKay conjecture to a question on finite simple groups. This is joint work with G. Navarro and B. Späth.