Let $p$ be a prime and $q$ be a power of $p$. Let $k$ be an algebraic closure of $\mathbb F_q$, the field of $q$ elements. Let $N$ be an integer $\geq 2$, and $G = GL_N(k)$. There is an involutive automorphism $$\theta: G \rightarrow G,$$ sending $g \in G$ to $J^{-1}(^tg^{-1})J$, where $J \in G$ is given by $$J = \begin{cases} \begin{pmatrix} 0 & 1_n \\ 1_n & 0 \end{pmatrix} & \text{ if } N=2n, \\ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1_n \\ 0 & 1_n & 0 \end{pmatrix} & \text{ if } N=2n+1. \end{cases}$$ Then $H: = (G^{\theta})^0$, the identity component of the $\theta$-stable subgroup of $G$, is the special orthogonal group.
Let $$F: G \rightarrow G$$ be the Frobenius map, sending $(a_{ij}) \in G$ to $(a_{ij}^q) \in G$. Obviously, both $H$ and $G^{\theta}$ are $F$-stable.
The group of rational points $G^F = \{ g \in G| F(g) = g\} = GL_N(\mathbb F_q)$ is a finite group, containg $(G^\theta)^F$ and $H^F$ as subgroups.
The order of $GL_N(\mathbb F_q)$ is well-known. Then
What is the order of $(G^{\theta})^F$ and $H^F$?
I tried to calculate. But the calculation turned out to be more and more complicated when $n$ went up. I believe that there are existing results on the orders of these two groups, but I cann't find one. Thank you very much for the help.