Galois automorphisms and blocks covering unipotent blocks

In this paper we prove that a recent condition of Lyons–Martínez–Navarro–Tiep, regarding the field of values of extensions of characters in principal blocks, is satisfied for all finite simple groups, which when combined with their results gives a new characterization of finite groups with a normal $\mathrm{\ell }$-complement for a prime $\mathrm{\ell }$. This leads us to study the distribution of characters in unipotent blocks of disconnected reductive groups and show that this is well-behaved under a generalization of $d$-Harish-Chandra theory. We go on to study the blockwise Galois–McKay (also known as the Alperin–McKay–Navarro) conjecture for the blocks of almost (quasi-)simple groups above unipotent blocks.