The $C_2$-equivariant ordinary cohomology of complex quadrics II: The symmetric case

In this, the second of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth symmetric quadrics graded on the representation ring of $\mathrm{\Pi }BU(1)$ and with coefficients in the Burnside Mackey functor. These calculations exhibit various interesting properties, including the first naturally occurring example we are aware of where the cohomology is not just the sum of shifted copies of the cohomology of a point, but also has summands that are shifted copies of the cohomology of the free orbit $C_2\mathrm{/}e$.