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.

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