When are induction and conduction functors isomorphic ?

Let R = ⊕ g∈GRg be a G-graded ring. It is well known (see e.g. [D], [M1], [N], [NRV], [NV]) that in the study of the connections that may be established between the categories R-gr of graded R-modules and R1-mod (1 is the unit element of G), an important role is played by the following system of functors : (−)1 : R-gr → R1-mod given by M 7→ M1, where M = ⊕ g∈GMg is a graded left R-module, the induced functor, Ind : R1-mod→ R-gr, which is defined as follows : if N ∈ R1mod, then Ind(N) = R⊗R1N which has the G-grading given by (R ⊗R1 N)g = Rg ⊗R1 N, ∀g ∈ G, and the coinduced functor, Coind : R1-mod→ R-gr, which is defined in the following way : if N ∈ R1-mod, then Coind(N) = ⊕ g∈GCoind(N)g , where Coind(N)g = {f ∈ HomR1(R1RR, N) | f(Rh) = 0, ∀ h 6= g−1} .