Condition : Let $G \le S_n$ acts on set $[n]$ and the action of $G$ on $[n]$ is $A_n$ i.e. it induce an homomorphism $\phi : G \mapsto S_n$, where image$(\phi) = A_n$.
Prove or Disprove: If $G$ satisfy the above written condition then maximal subgroup of $G$ is unique ( there is one only ).
This how I am thinking : Every group action on cosets is block system (each coset is a block ), now for a minimal block system coset size should be small (that means maximal subgroup). The above defined group action gives unique minimal block system, so there has to be one maximal subgroup.
Any high level idea is also fine.
Take $G = A_n$ and the conditions are satisfied. But $A_n$ doesn't have a unique maximal subgroup. In particular as a counterexample you can take $A_4$ and show that it has maximal subgroups of order $4$ and $3$.