Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let

$$ O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\} $$

What is $$ H^*(BO(\mathbb{Z}_2^{\oplus k});\mathbb{Z})? $$ If it cannot be computed out, can we get $$ H^*(O(\mathbb{Z}_2^{\oplus k});\mathbb{Z}_2)? $$ or for prime $p\geq 3$, $$ H^*(O(\mathbb{Z}_2^{\oplus k});\mathbb{Z}_p)? $$