Glauberman-Watanabe corresponding $p$-blocks of finite groups with normal defect groups are Morita equivalent

Abstract

Let $G$ be a finite group and let $A$ be a solvable finite group that acts on $G$ such that the orders of $G$ and $A$ are relatively prime. Let $b$ be a $p$-block of $G$ with normal defect group $D$ such that $A$ stabilizes $b$ and $D\leq C_{G}(A)$. Then there is a Morita equivalence between the block $b$ and its Watanabe correspondent block $W(b)$ of $C_{G}(A)$ given by a bimodule $M$ with vertex $\Delta D$ and trivial source that on the character level induces the Glauberman correspondence (and which is an isotypy by a theorem of Watanabe).