Let $G \leq S\_n$ be $2$-transitive other than $A\_n$ and $S\_n$. Is it possible that there exists $N\lhd G$ with $N\neq G$, $N$ transitive and $G/N$ cyclic? 


I am interested mostly in the answer when $n$ is large and also when the group $G$ is $3$-transitive. 



Is there a 2 transitive finite group with transitive normal subgroup having a cyclic quotient other than $A_n$ and $S_n$?
For $G = \operatorname{Aut}(M\_{22})$ and $N = M\_{22}$, with the action of $M\_{22}$ on $22$ points you have $N \triangleleft G < S\_{22}$. Here both $N$ and $G$ are $3$-transitive, and $G/N \cong C\_2$.


