Poincaré duality in P.A. Smith theory

Let $G=S^1$, $G=\mathbb{Z}/p$ or more generally $G$ be a finite $p$-group, where $p$ is an odd prime. If $G$ acts on a space whose cohomology ring fulfills Poincaré duality (with appropriate coefficients $k$), we prove a mod $4$ congruence between the total Betti number of $X^G$ and a number which depends only on the $k[G]$-module structure of $H^*(X;k)$. This improves the well known mod $2$ congruences that hold for actions on general spaces.