The $\mathbb{Z}\mathrm{/}p$ ordinary cohomology of $B_{G}U(1)$

With $G=\mathbb{Z}\mathrm{/}p$, $p$ prime, we calculate the ordinary $G$-cohomology (with Burnside ring coefficients) of $\mathbb{C}{P}_G^{\mathrm{\infty }}=B_{G}U(1)$, the complex projective space, a model for the classifying space for $G$-equivariant complex line bundles. The $RO(G)$-graded ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators, including the Euler class of the canonical bundle, as well as a significantly simpler set of relations.