Characterizing normal Hall subgroups by character degrees, II

We prove the character-degree assertion of Liu, Yang and Zhang for every proper nontrivial Hall subgroup of a finite simple classical group. Together with their reduction, this proves their normality criterion for Hall subgroups of arbitrary finite groups: a Hall $\pi$-subgroup $H$ of a finite group $G$ is normal if and only if every irreducible constituent of $1_H^G$ has ${\pi }^{\mathrm{\prime }}$-degree.