Endomorphism structure of Klein four-group

This article is about the structure of endomorphisms (and in particular automorphisms) of the Klein four-group, which is the elementary abelian group of order 4, or equivalently, the direct product of two copies of cyclic group:Z2.

Description of endomorphism monoid
The Klein four-group can be viewed as a two-dimensional vector space over field:F2. Moreover, endomorphisms of this as a group are precisely the same as $$\mathbb{F}_2$$-linear maps from this vector space to itself. These endomorphisms are described as $$2 \times 2$$ matrices over $$\mathbb{F}_2$$, with endomorphism composition given by matrix multiplication. Note that this identification depends on a choice of basis for the group as a vector space over $$\mathbb{F}_2$$.

Below is the complete list of endomorphisms, grouped together by similarity type of matrices (which means by conjugacy via automorphisms):