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][1] [article](https://doi.org/10.1016/0040-9383(65)90001-7 "The cohomology of quotients of classical groups"). 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?


  [1]: http://www.ams.org/mathscinet-getitem?mr=189063
  [2]: https://www.sciencedirect.com/science/article/pii/0040938365900017