Let $Ss(4m)$ be the $Z/2Z$ quotient of $Spin(4m)$ which is not $SO(4m)$. Its $Z/2Z$ cohomology was determined e.g. by Baum and Browder MR article. Is the $Z/2Z$ cohomology of its classifying space determined somewhere? What is \begin{equation} H^*(BSs(4m),Z/2Z) ? \end{equation}