Glauberman Z*-theorem

Statement
Let $$G$$ be a finite group. Denote by $$Z^*(G)$$ the normal subgroup containing $$O(G)$$ (the Brauer core) such that $$Z^*(G)/O(G) = Z(G/O(G))$$.

Let $$S$$ be a 2-Sylow subgroup of $$G$$. Suppose $$S$$ contains an involution $$t$$ which is not conjugate (in $$G$$) to any other element of $$S$$. Then $$t \in Z^*(G)$$. In particular, the existence of such an involution implies that $$G$$ is not a simple group.