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We prove that for most groups of Lie type, the bijections used by Malle andSpaeth in the proof of Isaacs-Malle-Navarro's inductive McKay conditions forthe prime 2 and odd primes dividing q - 1 are also equivariant with respect tocertain Galois automorphisms. In particular, this shows that these bijectionsare candidates for proving Navarro-Spaeth-Vallejo's recently-posited inductiveGalois-McKay conditions. On the way, we show that several simple groups of Lietype satisfy the McKay--Navarro conjecture for the prime 2.

Galois-equivariant McKay bijections for primes dividing $q-1$