Is a finite group action on a finite set determined by its fixed points?

If $G$ acts transitively on $X$, and $x_{0}$ is a fixed element of $X$, your information will yield the stabilizer $G_{x_{0}}$, as the largest subgroup $H$ such that $x_{0} \in X^{H}$. So you will know that the action of $G$ on $X$ is isomorphic to that of $G$ on the cosets of $G_{x_{0}}$.