On the Glauberman and Watanabe correspondences for blocks of finite $p$-solvable groups

If $G$ is a finite $p$-solvable group for some prime $p$, $A$ a solvable subgroup of the automorphism group of $G$ of order prime to $\vert G\vert$such that $A$ stabilises a $p$-block $b$ of $G$ and acts trivially on a defect group $P$ of $b$, then there is a Morita equivalence between the block $b$ and its Watanabe correspondent $w(b)$ of $C_{G}(A)$, given by a bimodule $M$ with vertex $\Delta P$ and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabe's results).