$\DeclareMathOperator\SL{SL}$Let $q = p^r$ be a prime power. Let $H$ denote the subgroup of $\SL(2,\overline{\mathbb{F}}_q)$ consisting of matrices of the form $\begin{pmatrix}a & b\\ b^q & a^q\end{pmatrix}$, where $a,b\in\mathbb{F}_{q^2}$. Some notes I'm reading claim that $H$ is conjugate to $\SL(2,\mathbb{F}_q)$.

This is a very naive question — How does one classify the conjugates of $\SL(2,\mathbb{F}_q)$ inside $\SL(2,\overline{\mathbb{F}}_q)$? How do you generally classify the conjugates of $G(\mathbb{F}_q)$ inside $G(\overline{\mathbb{F}}_q)$ when $G$ is say a split reductive algebraic group over $\mathbb{F}_p$?

Also, since I expect this is easy to answer for the experts here, what is the explicit matrix that conjugates $H$ to $\SL(2,\mathbb{F}_q)$?

References would be appreciated!

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