Endomorphism structure of special linear group of degree two over a finite field

Automorphism structure
For any prime power $$q$$, the automorphism group of the special linear group of degree two $$SL(2,q)$$ over the finite field $$\mathbb{F}_q$$ is the projective semilinear group of degree two $$P\Gamma L(2,q)$$.

Let $$q = p^r$$ where $$p$$ is the underlying prime. The information is presented below:

Other endomorphisms
If $$q$$ is 4 or more, SL(2,q) is quasisimple. Further, we have that finite quasisimple implies every endomorphism is trivial or an automorphism. Combining, we get that for $$q \ge 4$$, the endomorphisms of $$SL(2,q)$$ are the automorphisms and the trivial endomorphism.

The cases $$q = 2$$ (giving symmetric group:S3 -- see endomorphism structure of symmetric group:S3) or $$q = 3$$ (giving special linear group:SL(2,3) -- see endomorphism structure of special linear group:SL(2,3)) are somewhat different.

Facts about endomorphism structure

 * Transpose-inverse map is inner automorphism on special linear group of degree two
 * Special linear group of degree two has a class-inverting automorphism
 * Special linear group of degree two is ambivalent iff -1 is a square