Maximal subgroups of order $pq^2$ in finite simple groups

Question

This question is moved from math stackexchange, seems like it is a more advanced question. Here the link from the original question: https://math.stackexchange.com/questions/3415338/maximal-subgroups-of-order-pq2-in-finite-simple-groups

It is a well-known result in elementary group theory that if $q^{2}\mid (p-1)$, then there are two non-isomorphic nonabelian groups of the form $\mathbb{Z}_{q^{2}}\ltimes\mathbb{Z}_{p}$. One has a cyclic subgroup of order $pq$ while in the other one the subgroup of order $pq$ is nonabelian. Now my question:

Is there any finite simple group $G$ with a maximal subgroup $M\cong\mathbb{Z}_{q^{2}}\ltimes\mathbb{Z}_{p}$ such that the subgroup of order $pq$ in $M$ is abelian?

As far as I searched in ATLAS there is not such an example but I need a proof.

Answer 1

No there is no such simple group.

Suppose that $G$ is simple and has $M$ as a maximal subgroup. Let $P \in {\rm Syl}_q(M)$ and $Z=Z(M)$. So $|P|=q^2$ and $|Z|=q$ with $Z < P$. Since $M$ is maximal in $G$ and $M$ is not normal in $G$, we have $M = N_G(Z)$.

Now $N_G(Z)$ contains the centre of a Sylow $q$-subgroup $R$ of $G$ containing $P$, and since $Z$ is the only subgroup of $P$ of order $q$, we must have $Z \le Z(R)$, and so $R=P$.

Also, $N_G(P) \le N_G(Z) = M$, so $N_G(P) = N_M(P) = P$. But now by Burnside's Transfer Theorem $G$ has a normal $q$-complement in $G$, contradicting its simplicity.