Let $Ss(4m)$ be the $Z/2Z$ quotient of $Spin(4m)$ which is not $SO(4m)$. (This group is sometimes called the semi-spin group.) 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}

Update: In the string theory application I have in mind, it would be enough to know it up to degree 11. Does this make the determination any easier?