Let $X$ be a set with $p$ elements, where $p$ is prime. Also let $G$ be a subgroup of $S_X$ such that $|G|$ divides $p-1$. Moreover, there is an element $a\in X$ such that $g$ fixes $a$ for all $g\in G$. Is it always possible to put a group structure on $X$ so that $G\subseteq Aut(X)$?
More formally,
Suppose $X=\{0,1,2,\ldots,p-1\}$, where $p$ is prime. $G$ is a subgroup of $S_X$ such that $|G|$ divides $p-1$ and $g(0)=0$ for all $g\in G$. Is there always a bijection $f:X\to\mathbb{Z}_p$ such that $f\circ g\circ f^{-1}\in Aut(G)$?