Given a group $$ G= PSU(N) \rtimes \mathbb{Z}_2, $$ where $PSU(N)$ is a projective special unitary group. Say $a \in PSU(N)$, $c \in \mathbb{Z}_2$, then $$ c a c= a^*, $$ which $c$ flips $a$ to its complex conjugation written as $a^*$.
My question is what can we say about the group cohomology of $G= PSU(N) \rtimes \mathbb{Z}_2$ with integer coefficient or finite abelian group coefficient? Say $$ H^k(G, \mathbb{Z})=? $$ $$ H^k(G, \mathbb{Z}_2)=? $$ $$ H^k(G, \mathbb{Z}_n)=? $$ Here we can either regard the group cohomology $H^k(G,*)$, or regard it as the topological cohomology of the classifying space of the group $H^k(BG,*)$.
I am ONLY interested in $k=1,2,3,4$. Here $n=2$ or $n=4$ is enough. And when $N$ as an even integer is enough.
P.S: For your assitance --- When $N=2$, I already know that $$ H^3(SO(3), \mathbb{Z})=H^2(SO(3), \mathbb{R}/\mathbb{Z})=H^2(SO(3), \mathbb{Z}_2)=\mathbb{Z}_2. $$ $$ H^4(SO(3), \mathbb{Z})=H^3(SO(3), \mathbb{R}/\mathbb{Z})=\mathbb{Z}, $$ $$ H^2(PSU(N), \mathbb{Z}_N)=\mathbb{Z}_N, $$
See a related post: Group cohomology of orthogonal groups with integer coefficient
