  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 $n>1$.
