On the relative character correspondences

Let π(G,A):IrrA(G)→Irr(CG(A)) be the Glauberman–Isaacs correspondence, where G and A are finite groups with coprime orders and A acts on G by automorphisms. Let B be a subgroup of A. In this setting, we give some new conditions for the fixed-point subgroups CG(A) and CG(B) such that χπ(G,A) is an irreducible constituent of the restriction of χπ(G,B) to CG(A) for all χ IrrA(G).