Glauberman correspondence

Definition
Suppose $$S$$ is a finite solvable group acting on a finite group $$G$$. Assume that the orders of $$S$$ and $$G$$ are relatively prime. Then, the Glauberman correspondence is a natural bijection between the $$S$$-invariant irreducible linear characters of $$G$$, and the $$S$$-invariant irreducible linear characters of $$C_G(S)$$ (the fixed points in $$G$$ under the $$S$$-action).