Principal 2-Blocks and Sylow 2-Subgroups

Let G be a finite group with Sylow 2-subgroup P 6 G. Navarro–Tiep–Vallejo have conjectured that the principal 2-block of NG(P) contains exactly one irreducible Brauer character if and only if all odd-degree ordinary irreducible characters in the principal 2-block of G are fixed by a certain Galois automorphism σ. By recent work of Navarro–Vallejo it suffices to show this conjecture holds for all almost simple groups whose socle has 2-power index. In this paper we show that this conjecture holds for all such groups except when the socle is of type Bn(q), Cn(q), Dn (q), or E8(q) with q odd. Finally we show that in these outstanding cases it is sufficient to show that the finite simple group is SN2S-Good, in the sense of [SF16, Definition 1].