Let $G$ be an almost simple group with socle $PSL(2,q)$ where $q=p^f>3$ is the $f$th power of some odd prime $p$, and $M$ a maximal subgroup of $G$. By http://arxiv.org/abs/math/0703685, except for some small cases, $M\cap PSL(2,q)$ is maximal in $PSL(2,q)$ whose subgroups are explicitly known. Denote by $M(G)$ the maximal subgroup of $G$ such that $M(G)\cap PSL(2,q)=D_{q+1}$.
How many conjugacy classes are there for $M(G)$ in $G$?
Does anyone know more about the structure of $M(G)$ as an abstract group than $M(G)$ is just an extension of $D_{q+1}$ by $Out(G)$? Expecially for $G=P\Sigma L(2,q)$ and $P\Gamma L(2,q)$ when $M(P\Sigma L(2,q))$ and $M(P\Gamma L(2,q))$ are shown to be $N_G(D_{q+1})$ and $N_G(D_{2(q+1)})$ respectively in http://arxiv.org/abs/math/0703685.
I'm working with the transitivity of the natural action of $M(G)$ on $PG(1,q)$. It is not hard to prove that $M(G)$ is transitive on $PG(1,q)$ for $G\geq PGL(2,q)$, and $M(PSL(2,q))$ is transitive on $PG(1,q)$ iff $q\equiv3\pmod{4}$. Based on computation results, I think that $M(P\Sigma L(2,q))$ is transitive on $PG(1,q)$ iff $p\equiv3\pmod{4}$. Then how to prove this and what about the remaining cases of $G$?