Character Correspondences and Subgroups of Operator Groups

LetAandGbe finite groups with coprime orders, and suppose thatAacts onGby automorphisms. Let π(G, A) : IrrA(G) → Irr(CG(A)) be the Glauberman–Isaacs correspondence. LetB ≤ Aand let χ IrrA(G). We examine the conjecture that χπ(G, A) is an irreducible constituent of the restriction of χπ(G, B) to CG(A) and show that it is valid ifGis supersolvable. Then, we show when the analog of this conjecture for Brauer characters and the Uno–Wolf correspondence holds.