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

This page describes the endomorphism structure of $$PGL(2,q)$$, the projective general linear group of degree two over a finite field of size a prime power $$q$$. We denote by $$p$$ the underlying prime of $$q$$, so $$q = p^r$$ where $$r = \log_pq$$.