Let $G$ be a finite group and $\Sigma$ a homology sphere with smooth $G$-action. If the $G$-fixed-point set of $\Sigma$ consists of odd-number points then the dimension of $\Sigma$ could be restrictive. In this article we confirm the claim in the cases where $G = S_5$ or $S\!L(2, 5)$.
"Spheres not admitting smooth odd-fixed-point actions of $S_5$ and $SL(2, 5)$." Osaka J. Math. 57 (1) 1 - 8, January 2020.