  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 of G as an object under the stable p-completed classifying
space of S 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
p-completed homotopy type of BG is determined by the stable p-completed
inclusion of BS in BG, but not by the stable p-completed homotopy type of BG.
