Characters of relative π′π‐degree over normal subgroups

The Gluck–Wolf theorem and its general version [Navarro and Tiep, Annals of Math. 178 (2013), 1135–1171] relate arithmetic properties at a fixed prime of the ratios , for irreducible characters of a finite group that lie over a fixed ‐invariant irreducible character of a normal subgroup of , to the structure of Sylow ‐subgroups of . This result constituted a key step towards the recent proof [Malle et al., Annals of Math. 200 (2024), 557–608] of Brauer's Height Zero Conjecture. In this paper, we prove a further extension of the Gluck–Wolf theorem to sets of primes, with a mild condition on if the alternating group is involved in the group.