Computations of de Rham cohomology rings of classifying stacks at torsion primes

For the split group $G_2$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B(G_2{)}_{{\mathbb{F}}_2}$ is isomorphic to the singular cohomology ring with ${\mathbb{F}}_2$-coefficients of $B(G_2{)}_{\mathbb{C}}\mathrm{.}$ For the spin groups $\text{Spin}(n)$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B\text{Spin}(n{)}_{{\mathbb{F}}_2}$ is isomorphic to the singular cohomology ring with ${\mathbb{F}}_2$-coefficients of $B\text{Spin}(n{)}_{\mathbb{C}}$ for $n\le 10.$ For $n=11,$ we make a full computation of the de Rham cohomology ring of $B\text{Spin}(11{)}_{{\mathbb{F}}_2},$ which is not isomorphic to the singular cohomology ring with ${\mathbb{F}}_2$-coefficients of $B\text{Spin}(11{)}_{\mathbb{C}}\mathrm{.}$ We also show that the Hodge spectral sequence for $B{G}_{{\mathbb{F}}_2}$ degenerates for all of the groups $G$ mentioned above.