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 ${\Sigma }^{\infty }{\mathrm{BG}}_p^{\wedge }$ as an object under ${\Sigma }^{\infty }\mathrm{BS}$ 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 ${\mathrm{BG}}_p^{\wedge }$ is determined by the map ${\Sigma }^{\infty }\mathrm{BS}\to {\Sigma }^{\infty }{\mathrm{BG}}_p^{\wedge }$, but not by the homotopy type of ${\Sigma }^{\infty }{\mathrm{BG}}_p^{\wedge }$.