Let $p\geq 5$ be a prime and $\mathbf{F}_p$ a finite field of characteristic $p$. A subgroup of ${\rm GL}_n(\mathbf{F}_p)$ is full if it contains ${\rm{SL}}_n(\mathbf{F}_p)$. When $n=2$, we have the following nice characterisation of full subgroups, cf. P.182,Theorem 2.1 in [Lang, S., 2001. Introduction to modular forms (Vol. 222).]

Let $G$ be a subgroup of ${\rm GL}_2(\mathbf{F}_p)$. Then $G$ is a full subgroup if and only if the order of $G$ is divisible by $p$ and $G$ is non-solvable.

My question is following:

Let $G$ be a subgroup of ${\rm GL}_n(\mathbf{F}_p)$ for some $n\geq3$. Assume that the order of $G$ is divisible by $p$ and $G$ is non-solvable. Is it possible to add some slight conditions to ensure that $G$ is a full subgroup?