I have a questions regarding an application of the theorem of Schur-Zassenhaus and/or Frattini's argument. I am trying to figure out the following:
Let $p\geq n$ and $G<\text{GL}_n(\mathbb{F}_q)$ be a subgroup, where $\mathbb{F}_q$ is a finite extension of $\mathbb{F}_p$. Let $P<G$ be a $p$-Sylow subgroup and $G^+<G$ the subgroup of $G$ generated by the elements $g\in G$ with $g^p=1$. We know that $p \nmid[G:G^+]$.
Now I want to deduce, that $$G= G^+ N_G(P),$$ where $N_G(P)$ denotes the normalizer of $P$ in $G$. I thought of using Frattini's argument, which however only works if $G^+$ is normal. Does anyone know, how to deduce the general case? The fact $p\nmid [G:G^+]$ looks like Schur-Zassenhaus, but it is not directly applicable and I don't know how to do it.
Thank in advance for any hints.