Canonicity of the refined equivalence

Ascertain whether the equivalence between the F(Γ;Z)-representation V_Z(Γ,G) and the F(Γ;Z^∧)-representation V_{Z^∧}(Γ,\tilde H) predicted in the refined conjecture can be defined canonically, in light of the fact that the action ρ of F(Γ;Z) on V_Z(Γ,G) is equivalent to ρ∘inv, where inv sends each element a∈F(Γ;Z) to a^{-1}.

Background

In discussing the refined conjecture, the authors note a subtlety: the group action on V_Z(Γ,G) admits an automorphism inv that inverts elements of F(Γ;Z), and the resulting actions ρ and ρ∘inv are equivalent. This raises a question about whether the predicted equivalence with V_{Z^∧}(Γ,\tilde H) is canonical or depends on choices.

Clarifying this canonicity would strengthen the refined conjecture by specifying whether a distinguished, natural isomorphism exists between the two representations, or only a non-canonical equivalence class.