We study OMEGA(n)Spin,Z(p), n = 5, 6, 7, where p is an odd prime and we consider some applications to the existence of Z(p)-invariant metrics of positive scalar curvature on closed spin manifolds of dimensions 5, 6, 7. We show that for actions of "mixed type" the vanishing of the obstruction considered implies Z(p)-invariant positive scalar curvature.