Representations of the alternating group which are irreducible over subgroups, II

We prove that non-trivial representations of the alternating group ${\mathsf A}_n$ are reducible over a primitive proper subgroup which is isomorphic to some alternating group ${\mathsf A}_m$. A similar result is established for finite simple classical groups embedded in ${\mathsf A}_n$ via their standard rank $3$ permutation representations.