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For a prime $\ell$, the McKay conjecture suggests a bijection between the setof irreducible characters of a finite group with $\ell'$-degree and thecorresponding set for the normalizer of a Sylow $\ell$- subgroup. Navarro'srefinement suggests that the values of the characters on either side of thisbijection should also be related, proposing that the bijection commutes withcertain Galois automorphisms. Recently, Navarro--Spaeth--Vallejo have reducedthe McKay--Navarro conjecture to certain "inductive" conditions on finitesimple groups. We prove that these inductive McKay--Navarro (also called theinductive Galois--McKay) conditions hold for the prime $\ell = 2$ for severalgroups of Lie type, including the untwisted groups without nontrivial graphautomorphisms.

The inductive McKay-Navarro conditions for the prime 2 and some groups of Lie type