Given $G = PSL(2,p^2)$ ($p$ prime), I wish to know what are the possible conditions such that any two maximal subgroups of $G$ that are isomorphic to $PGL(2,p)$, their intersection will contain either a form of $PGL(2,q_1)$ (some $q_1$), $A_5$ or $S_4$ (assuming $q$ is large).
This question is a follow-up question of this. Therefore, I am aware of the counter examples, thus, wish to know if there are certain constraints that make the desired intersection possible.
Any comments or suggestions are greatly appreciated.