Correspondences of isomorphism types of irreducible modules in finite group block theory

Let k be an algebraically closed field of prime characteristic p>1, let G be a finite group, let and let B be a block of kM with defect group P. Set H=NG(P) so that B⁎=BrP(B) is a block of k(H∩M) with defect group P. Assume that M is p-solvable. We prove that there is a StabH(B)-bijection, φ(B,P), between the isomorphism types of irreducible k(H∩M)B⁎-modules with vertex P and (kM)B-modules with vertex P. Moreover, for each block L of G that covers B, so that L⁎=BrP(L) is a block of kH, there is a bijection between isomorphism types of irreducible (kH)L⁎-modules and (kG)L-modules that “cover” the isomorphic module types in the domain and target of φ(B,P) that is “consistent” with φ(B,P) and preserves vertices.