Finite simple groups with three conjugacy classes of maximal local subgroups $\DeclareMathOperator\PSL{PSL}$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)
 A: If you want to use CFSG, I think this is doable (and may even be doable without CFSG if you use H. Bender's classification of finite groups with a strongly embedded subgroup, with some additional work).
For sporadic groups, is a matter of checking. In an alternating group $G$, there are three non-conjugate maximal local subgroups,
$N_{G}(\langle (123) \rangle)$, $N_{G}( \langle (12)(34), (13)(24) \rangle )$ and $N_{G}(\langle (12345) \rangle)$, and for $n \geq 7$, it is easy to construct maximal local subgroups not conjugate to any of these.
For simple groups of Lie type of defining characteristic $p$, then for rank at last three, there are at least three conjugacy classes of maximal $p$-locals ( which are parabolics here) which are also non-conjugate maximal local subgroups.
Also, (with a few exceptions), the normalizer of the maximal torus ,$T$, of the Borel is contained in a maximal local subgroup which is not conjugate to any parabolic.
Hence the real work is in dealing with simple groups of Lie type of defining characteristic $p$ and of rank at most $2$, and this should be manageable.
