Is maximal subgroup unique?

Question

Claim: If there is an homomorphism $\phi$ from $G$ (finite group) to $S_n$ and $G/\ker (\phi) \cong A_n $ then the maximal subgroup of $G$ is unique.

Question: Is the above claim true?

Answer 1

A conjugate of a maximal subgroup is again maximal. Thus if the maximal subgroup is unique, it is normal. (And the quotient is some $\mathbb Z/p$.)

$A_n$ has no nontrivial normal subgroups for $n\geq5$, so the inclusion of $A_n$ gives a counter-example.

Answer 2

It is not restrictive to assume that the image of $\phi$ is $A_n$ (the unique subgroup of $S_n$ of index $2$).

Let $M$ be a maximal subgroup of $A_n$, which has several of them when $n\ge5$ (why?). Then $\phi^{-1}(M)$ is a maximal subgroup of $G$ and the correspondence theorem says that distinct subgroups of $A_n$ have distinct inverse images.