$\DeclareMathOperator\PSL{PSL}$In [1] it was proved that

A finite nonsolvable group $G$ has three conjugacy classes of maximal subgroups if and only if $G/\Phi(G)$ is isomorphic to $\PSL(2,7)$ or $\PSL(2,2^q)$ for some prime $q$. This implies that, among finite simple groups, only only $\PSL(2,7)$ and $\PSL(2,2^q)$ have three conjugacy classes of maximal subgroups.

**My question:** I wonder if we can also find all finite simple groups with three conjugacy classes of * maximal local subgroups*.

A subgroup is a *local subgroup* if it is the normalizer of some nontrivial subgroup of prime power order. A proper local subgroup is a *maximal local subgroup* if it is maximal among proper local subgroups.

Maximal subgroups are not necessarily local, and maximal local subgroups are not necessarily maximal subgroups. I know that the three non-conjugate maximal subgroups of $\PSL(2,4)=A_5$ and $\PSL(2,7)$ are local respectively, but is it true that $\PSL(2,2^q)$ has three conjugacy classes of maximal local subgroups for each prime $q$? And how can I find all simple groups with such property?

Any help is appreciated!

**Reference:**

[1] Belonogov, V. A.: Finite groups with three classes of maximal subgroups. Math. Sb., 131, 225–239 (1986)