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], w) are equivariant-surgery-obstruction groups and showed in the case of even dimension n that the Bak-group functor W_n(Z)[–], _–; 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.