Abstract: We prove that, confined that G > H > P and P is a proper p-subgroup of H, if H ∩~gH ≤ P for any g ∈ G-H, then the operator of the restriction to RH of RG-modules induces a triangulated equivalence from StmodP(RG) to StmodP(RH); if the normal subgroup H controls the fusion of p-subgroups of G, the restriction functor is a faithful triangulated functor; if P is strongly closed in H respect to G, the same functor is also a faithful triangulated functor. 

Key words: stable module category, restriction functor, category equivalence