Mackey and Frobenius structures on odd dimensional surgery obstruction groups

C. T. C. Wall formulated surgery-obstruction groups L_n(ℤ[G] ) in terms of quadratic modules and automorphisms. C. B. Thomas showed that the Wall-group functors L_n(ℤ[-], w|_) (are modules over the Hermitian-representation-ring functor G₁(ℤ-) if the orientation homomorphism w is trivial. A. Bak generalized the notion of quadratic module by introducing quadratic-form parameters, and obtained various K-groups related to quadratic modules and automorphisms. One of the authors established that some Bak groups W_n(ℤ[G], Λ; w) are equivariant-surgery- obstruction groups and showed in the case of even dimension n that the Bak-group functor W_n(ℤ[-], Λ_; w|_) is a w-Mackey functor as well as a module over the Grothendieck-Witt-ring functor GW₀(ℤ,-), where w is possibly nontrivial. In this paper, we prove the same facts in the case of odd dimension n.