SCHWEITZER 2015 Problem 6

Let $G$ be the permutation group of a finite set $\Omega$ .Consider $S\subset G$ such that $1\in S$ and for any $x,y\in \Omega$ there exists a unique element $\sigma \in S$ such that $\sigma (x)=y$ .Prove that,if the elements of $S \setminus \{1\}$ are conjugate in $G$ ,then $G$ is $2-$ transitive on $\Omega$