$\DeclareMathOperator\BPSO{BPSO}\DeclareMathOperator\PSO{PSO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\BSO{BSO}$The projective group is $\PSO(2d)=\SO(2d)/Z_2$.

$\BPSO(2d)$ is the classifying space of $\PSO(2d)$.

Question: What is the cohomology of $\BPSO(2d)$ with $Z_2$ coefficients?

Note: Here we already know the cohomology of $\BSO(2d)$ with $Z_2$ coefficients. $$H^*(\BSO(2d),Z_2)=Z_2[w_2,w_3,...,w_{2d}].$$ Can we obtain that of $\BPSO(2d)$?