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 \G\ 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 DeltaP 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).