Noncoprime action and character correspondences (Q1324960)

If \(G\) is a complemented normal subgroup of a finite group \(\Gamma\) and \(C\) is a set of representatives of \(G\)-conjugacy classes of complements of \(G\) in \(\Gamma\), the author shows that there exists a natural map from some subset of the \(\Gamma\)-invariant characters of \(G\) (those who have \(p\)-defect zero for the primes dividing \(|\Gamma/ G|\)) into \(\bigcup_{S\in C} \text{Irr} (C_G (S))\), whenever \(G\) is \(\pi\)-separable for the set of primes \(\pi\) dividing \(|\Gamma/ G|\). This generalizes Nagao's extension of the Glauberman correspondence.