Invariant constituents and invariant blocks under coprime action

Let A and G be finite groups with (A,G)=1. We assume that A acts on G via automorphism. Let N be an A-invariant normal subgroup of G. Let be an A-invariant irreducible Brauer character of N. If A is of prime power order, then the induced Brauer character G contains an A-invariant irreducible constituent; If G/N is p-solvable, then G contains an A-invariant irreducible constituent. Let B be an A-invariant block of G. Then under Glauberman–Isaacs correspondence, the set IrrA(B) is a union of blocks of CG(A), say b1,b2,…,bs. Let Qi be a defect group of bi. Then there is a defect group D of B such that Qi D.