Endomorphism structure of linear groups of degree two

This page provides a summary, with links to more details, for the endomorphism structure of the general linear group of degree two and some other related groups, both over fields and over related rings.

Formulas
In the formulas below, the field size is $$q$$. The characteristic of the field is a prime number $$p$$. $$q$$ is a prime power with underlying prime $$p$$. We let $$r = \log_pq$$, so $$q = p^r$$ and $$r$$ is a nonnegative integer.