Alternative stable homotopy classification of BGp∧

We give an alternative to the stable classification of p-completed homotopy types of classifying spaces of finite groups offered by Martino–Priddy. For a finite group G with Sylow subgroup S, we regard the stable p  -completed classifying space Σ∞BGp∧ as an object under Σ∞BSΣ∞BS via the canonical inclusion map. Thus we get a classification in terms of induced fusion systems. Applying Oliver's solution to the Martino–Priddy conjecture, we obtain the surprising result that the unstable homotopy type of BGp∧ is determined by the map Σ∞BS→Σ∞BGp∧, but not by the homotopy type of Σ∞BGp∧.