  Following the Hu-Kriz method of computing the $C_2$ genuine dual Steenrod
algebra $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$, we calculate the $C_4$
equivariant Bredon cohomology of the classifying space $\mathbf R P^{\infty
\rho}=B_{C_4}\Sigma_{2}$ as an $RO(C_4)$ graded Green-functor. We prove that as
a module over the homology of a point (which we also compute), this cohomology
is not flat. As a result, it can't be used as a test module for obtaining
generators in $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$ as Hu-Kriz do in the
$C_2$ case.
