An old question of P. A. Smith considers a finite group G acting smoothly on a closed homotopy sphere \(\Sigma\) with \(\Sigma^ G\) consisting of precisely two points and asks whether the representations of G on the tangent spaces at the two fixed points are the same. This would be true, obviously, for a linear action on a sphere. For G abelian of odd order with at least four noncyclic Sylow subgroups this paper shows that the answer is no, providing proofs and elaboration of previously announced results. For a large family of representations one has necessary and sufficient conditions that two representations in the family occur as fixed point representation after stabilization.