The cohomology of $C_2$-equivariant Grassmannians

A database of known values of $H^{\ast,\ast}(Gr_k(\mathbb{R}^{p,q});\underline{\mathbb F}_2)$, expressed as a module over $\mathbb{M}_2:=H^{\ast,\ast}(pt;\underline{\mathbb F}_2)$. All $RO(C_2)$-grading is motivic, with bidegree $(x,y)$ indicating a $x$-dimensional representation containing $y$ copies of the sign representation (and hence $x-y$ copies of the trivial rep.)

Note that $Gr_k(\mathbb{R}^{p,p-q})\cong Gr_k(\mathbb{R}^{p,q})$ and $Gr_{p-k}(\mathbb{R}^{p,q})\cong Gr_k(\mathbb{R}^{p,q})$ so we'll just worry about when $k,q\le \frac p2$.

When $k=1$

These are the projective spaces $\mathbb{P}(Gr_k\mathbb{R}^{p,q})$ whose cohomologies were calculated by Kronholm, who defines ``twisted" projective space $\mathbb{RP}^n_{tw}=\mathbb{P}(\mathbb{R}^{n+1,\lfloor\frac n2\rfloor})$ and gives $$H^{\ast,\ast}(\mathbb{RP}^n_{tw})=\bigoplus_{k=0}^n\Sigma^{k,\lfloor\frac k2\rfloor}\mathbb M_2$$.
$k=1$ $q=1$ $q=2$ $q=3$ $q=4$ $q=5$
$p=2$ $\mathbb{RP}^1_{tw}$
$p=3$ $\mathbb{RP}^2_{tw}$
$p=4$ $\mathbb{P}(\mathbb{R}^{4,1})$ $\mathbb{RP}^3_{tw}$
$p=5$ $\mathbb{P}(\mathbb{R}^{5,1})$ $\mathbb{RP}^4_{tw}$
$p=6$ $\mathbb{P}(\mathbb{R}^{6,1})$ $\mathbb{P}(\mathbb{R}^{6,2})$ $\mathbb{RP}^5_{tw}$
$p=7$ $\mathbb{P}(\mathbb{R}^{7,1})$ $\mathbb{P}(\mathbb{R}^{7,2})$ $\mathbb{RP}^6_{tw}$
$p=8$ $\mathbb{P}(\mathbb{R}^{8,1})$ $\mathbb{P}(\mathbb{R}^{8,2})$ $\mathbb{P}(\mathbb{R}^{8,3})$ $\mathbb{RP}^7_{tw}$
$p=9$ $\mathbb{P}(\mathbb{R}^{9,1})$ $\mathbb{P}(\mathbb{R}^{9,2})$ $\mathbb{P}(\mathbb{R}^{9,3})$ $\mathbb{RP}^8_{tw}$
$p=10$ $\mathbb{P}(\mathbb{R}^{10,1})$ $\mathbb{P}(\mathbb{R}^{10,2})$ $\mathbb{P}(\mathbb{R}^{10,3})$ $\mathbb{P}(\mathbb{R}^{10,4})$ $\mathbb{RP}^9_{tw}$