A quotient of the set [BG,BU(n)] for a finite group G of small rank

Let Gp be a Sylowp-subgroup of the finite group G and let CharnG(Gp) represent the set of degree n complex characters of Gp that are the restrictions of class functions on G. We construct a natural map ψG : [BG,BU(n)]→∏pGCharnG(Gp) and prove that ψG is a surjection for all finite groups G that do not contain a subgroup isomorphic to for any prime p. We show, furthermore, that ψG is in fact a bijection for two types of finite groups G: those with periodic cohomology and those of odd order that do not contain a subgroup isomorphic to for any prime p.