TITLE: Computations in modular cohomology of finite groups
QUESTION [8 upvotes]: Let $k$ be an algebraically closed field of characteristic $p$, let $G$ be a finite group whose order is divisible by $p$, and let $H(G)$ be the commutative cohomology algebra of $G$ with coefficients in $k$ (viewed as a trivial module), i.e.,
$$H(G):=\begin{cases}H^*(G,k)&p=2\\H^{ev}(G,k)&p>2\end{cases}$$
For what sorts of groups have we explicitly computed $H(G)$?  Or if not an explicit computation, what can we say about $H(G)$ (or at least $|\operatorname{Spec}H(G)|$)?  Below is what I'm aware of:

If $G$ is elementary abelian of rank $r$, then $H(G)$ is known by using a particularly nice projective resolution of a cyclic group of order $p$ and using the Kunneth formula.
Thanks to Quillen, the Krull dimension of $H(G)$ is the $p$-rank of $G$, that is the length of the largest chain of prime ideals in $H(G)$ is equal to the largest rank of an elementary abelian $p$-subgroup of $G$.
Jon Carlson has used Magma to compute $H(G)$ for some $2$-groups here.

If you find the question too broad, at the moment I'm particularly interested in finite groups of Lie type, and to be even more specific, $\operatorname{GL}_n(\mathbb{F}_q)$.  I recall reading a paper that computed $H(\operatorname{GL}_2(\mathbb{F}_q))$, and I also recall reading another paper showing certain vanishing results of $H(\operatorname{GL}_n(\mathbb{F}_q))$ as one allows $n$ or $q$ to grow independently, but I can't seem to find either of these papers and I don't remember the authors.


What do we know about the commutative algebra $H(\operatorname{GL}_n(\mathbb{F}_q))$?  Have any computations been made for certain $n$ and $q$?


Any references are appreciated, even those that address the question for general $G$, and not just $\operatorname{GL}_n(\mathbb{F}_q)$.

REPLY [6 votes]: There is a comprehensive set of calculations of the homology of classical groups over finite fields in Z. Fiedorowicz and S. Priddy. Homology of classical groups over finite fields and their associated infinite loop spaces.  Springer Lecture Notes in Mathematics Volume 674, 1978. I'm a little surprised this has not yet been mentioned, since it seems to give more complete answers than some that are cited above.  It is written with a topological slant, following up Quillen's original work that used such homological calculations to determine the K-theory of finite fields, by comparison of Frobenius in algebra with Adams operations in topology.