Glauberman Correspondents And Blocks With Klein Four Defect Group

Let G bea finite group with abelian Sylow 2-subgroup, A a finite solvable group coprimly acting on G. Let GA be the subgroup of all A-stable elements of G. Let b be a block of G with Klein four defect group. Assume that A stabilizes the block b and centralizes the defect group of it. Then there is a Morita equivalence between the block b and its Glauberman correspondent ω(b) of GA.