$C_{2^n}$-equivariant rational stable stems and characteristic classes

In this short note, we compute the rational $C_{2^n}$-equivariant stable stems and give minimal presentations for the $RO(C_{2^n})$-graded Bredon cohomology of the equivariant classifying spaces $B_{C_{2^n}}S^1$ and $B_{C_{2^n}}\Sigma_2$ over the rational Burnside functor $A_{\mathbf Q}$. We also examine for which compact Lie groups $L$ the maximal torus inclusion $T\to L$ induces an isomorphism from $H^*_{C_{2^n}}(B_{C_{2^n}}L;A_{\mathbf Q})$ onto the fixed points of $H^*_{C_{2^n}}(B_{C_{2^n}}T;A_{\mathbf Q})$ under the Weyl group action. We prove that this holds for $L=U(m)$ and any $n,m\ge 1$ but does not hold for $L=SU(2)$ and $n>1$.