Power operations for $\text{H}{\underset{‾}{\mathbb{F}}}_2$ and a cellular construction of $\text{BP}\mathbf{R}$

We study some power operations for ordinary $C_2$-equivariant homology with coefficients in the constant Mackey functor ${\underset{‾}{\mathbb{F}}}_2$. In addition to a few foundational results, we calculate the action of these power operations on a $C_2$-equivariant dual Steenrod algebra. As an application, we give a cellular construction of the $C_2$ equivariant Brown-Peterson spectrum $\text{BP}\mathbf{R}$ and deduce its slice tower.