Nonexistence of smooth effective one fixed point actions of finite Oliver groups on low-dimensional spheres

According to Laitinen and Morimoto (1998), a finite group $G$ has a smooth effective one fixed point action on some sphere if and only if $G$ is an Oliver group. For some finite Oliver groups $G$ of order up to $216$, and for $G=A_5\times C_p$, where $p=3,5,7$, we present a strategy of excluding smooth effective one fixed point $G$-actions on low-dimensional spheres.