The cohomology of C2-equivariant 𝒜(1) and the homotopy of koC2

We compute the cohomology of the subalgebra $A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $A^{C_2}$. This serves as the input to the $C_2$-equivariant Adams spectral sequence converging to the $RO(C_2)$-graded homotopy groups of an equivariant spectrum $ko_{C_2}$. Our approach is to use simpler $\mathbb{C}$-motivic and $\mathbb{R}$-motivic calculations as stepping stones.