21.100 (2026)

Suppose that $A$ and $G$ are finite groups such that $A$ acts coprimely on $G$ by automorphisms. Let $C = C_G(A)$ be the fixed-point subgroup, and let $C'$ denote its derived subgroup. Is it true that the number of $A$-invariant irreducible characters $\chi$ of $G$ whose restriction $\chi_C$ is never zero is exactly $|C/C'|$?

This would follow if one could show that $\chi_C$ is never zero if and only if the Glauberman–Isaacs correspondent $\chi^*$ of $\chi$ is linear.