Richness of smith equivalent modules for finite gap oliver groups

Let G be a finite group not of prime power order. Two real G-modules U and V are T^GJ-connectively Smith equivalent if there exists a homotopy sphere with smooth G-Action such that the fixed point set by P is connected for all Sylow subgroups P of G, it has just two fixed points, and U and V are isomorphic to the tangential representations as real G-modules respectively. We study the P(G)-connective Smith set for a finite Oliver group G of the real representation ring consisting of all differences of P(G)-connectiveIy Smith equivalent G-modules, and determine this set for certain nonsolvable groups G.