I am reading the Kleidman-Liebeck book ("The subgroup structure of the finite classical groups") which is about the Aschbacher classification of maximal subgroups of the classical almost simple groups. But thinking just about natural actions of classical groups I got stuck in the following problem.

Consider the group $GL(n,q)$ of invertible $\mathbb{F}_q$-linear maps $\mathbb{F}_q^n \to \mathbb{F}_q^n$, where $n$ is a positive integer and $q=p^k$ is a prime power. This group acts on $V := \mathbb{F}_q^n$ in the natural way, and this action is $\mathbb{F}_q$-linear, in particular it is $\mathbb{F}_p$-linear. $\mathbb{F}_p$-linearity yields an injective homomorphism $GL(n,q) \to GL(nk,p)$, and I will identify $GL(n,q)$ with a subgroup of $GL(nk,p)$ by means of this homomorphism. Let $W$ be a proper $\mathbb{F}_p$-subspace of $V$, and let $H$ be the stabilizer of $W$ in $GL(nk,p)$, i.e. the set of invertible $\mathbb{F}_p$-linear transformations $f$ of $V$ such that $f(W)=W$.

Since $GL(n,q)$ acts transitively on non-zero vectors, $L := H \cap GL(n,q)$ is a proper subgroup of $GL(n,q)$, so it is contained in a maximal subgroup of $GL(n,q)$ which belongs to one of the Aschbacher classes. Which one?

If $W$ spans a proper $\mathbb{F}_q$-subspace of $V$ then $L$ is of parabolic type (class $\mathcal{C}_1$).

What happens if $\langle W \rangle_{\mathbb{F}_q} = V$? My guess is that in this case $L$ falls in class $\mathcal{C}_5$ (subfield stabilizers), but unfortunately I do not know enough about the linear groups to know how to think about this or where to look at.

I have been thinking about this and I asked people, but the question does not seem easy. This is just to say that I hope it is not trivial for experts, in which case I apologize.