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

Automorphism structure
For any prime power $$q$$, the automorphism group of the projective special linear group of degree two $$PSL(2,q)$$ 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, the group PSL(2,q) is simple, so the only endomorphisms are the trivial endomorphism and the automorphisms. If $$q = 2$$ (giving symmetric group:S3) or $$q = 3$$ (giving alternating group:A4) then there are other endomorphisms with nontrivial kernels.