Characters, coprime actions, and operator groups

Abstract.

In this paper, we answer the following question posed by G. Navarro: suppose that S acts on G coprimely, and that A acts on SG so that S and G are both A-invariant. If $\chi$ is an S-invariant irreducible character of G, then must $\chi$ extend to AG if and only if $\chi ^*$ extends to A C _G (S), where $\chi ^*$ is the Glauberman-Isaacs Correspondent for $\chi$ under the action of S? We prove that the answer to this question is yes.