Restriction of $p'$-degree characters, fields of values, and relative normal complements

Abstract

Let $G$ be a finite $p$-solvable group for a prime $p$, and let $P$ be a Sylow $p$-subgroup of $G$. Centella and Navarro proved that there is a canonical bijection ${\rm Irr}_{p'}(Φ_{1_G})\to{\rm Irr}_{p'}(Φ_{1_{\textbf N_G(P)}})$ preserving fields of values, where $Φ_{1_G}$ denotes the principal projective indecomposable character of $G$. In the spirit of a result of Berkovich on the McKay bijection, we characterize when this projective correspondence can be realized by restriction, proving that it holds if and only if $\mathbf{N}_G(P)$ admits a certain relative normal complement in $G$. More generally, we establish a refined criterion incorporating fields of values, and we extend this characterization to $π$-separable groups with nilpotent Hall $π$-subgroups.