Special unitary group of degree two equals special linear group of degree two over a finite field

Statement
Suppose $$q$$ is a prime power. Denote by $$SU(2,q)$$ the special unitary group corresponding to the quadratic extension $$\mathbb{F}_{q^2}$$ of the finite field $$\mathbb{F}_q$$. In particular, $$SU(2,q)$$ is the intersection of $$SL(2,q^2)$$ with $$U(2,q)$$. Then, $$SU(2,q)$$ is isomorphic to the special linear group of degree two $$SL(2,q)$$. Moreover, if we view both $$SU(2,q)$$ and $$SL(2,q)$$ as subgroups inside $$SL(2,q^2)$$, they are conjugate subgroups.

Related facts

 * Transpose-inverse map is inner automorphism on special linear group of degree two