Group actions on spheres with rank one prime power isotropy

We show that a rank two finite group G admits a finite G-CW-complex X similar or equal to S-n with rank one prime power isotropy if and only if G does not p'-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a given G-invariant family of representations defined on the Sylow subgroups of G.