Characters with stable irreducible constituents (Q1891492)

Let \(G\) be a finite group and \(A\) a group of automorphisms of \(G\) which acts on \(G\) coprimely. The author proves the following properties of ordinary irreducible \(A\)-invariant characters: (1) If \(H\) is an \(A\)-invariant subgroup of \(G\) and \(\chi\in \text{Irr} (G)\) is \(A\)-invariant then \(\chi_H\) contains some \(A\)-invariant irreducible constituent. (2) If \(H\) is an \(A\)-invariant subgroup of \(G\) and \(\theta\in \text{Irr} (H)\) is \(A\)-invariant, then \(\theta^G\) contains some \(A\)-invariant irreducible constituent. (3) If \(\alpha\) and \(\beta\) are irreducible \(A\)-invariant characters of \(G\), then \(\alpha \beta\) contains some \(A\)-invariant irreducible constituent.