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2 added 6 characters in body

I'm surprised nobody has mentioned

Here's one which is key for calculations: Let $H$ be a subgroup of $G$ and $W_G(H) = N_G(H)/H$. Then the restriction map $H^*(BG) \to H^*(BH)$ maps to the invariants $(H^*(BH))^{W_G(H)}$.

When $H$ is abelian, its cohomology is well-known (polynomial tensor exterior) and thus the cohomology of $G$ is mapping to something which can in principal be computed by invariant theory. Follow this with Quillen's theorem that the sum of these maps over all abelian subgroups has kernel which contains only nilpotent elements, and special cases such as Milgram's theorem that this is injective for symmetric groups, and you have a powerful computational tool.

Also, here is a nice survey by Alejandro Adem (whose book with Milgram is a good reference, complementary in many ways to Brown's). It is intended for a graduate student summer school audience: http://www.math.uic.edu/~bshipley/ConMcohomology1.pdf

I'm surprised nobody has mentioned: Let $H$ be a subgroup of $G$ and $W_G(H) = N_G(H)/H$. Then the restriction map $H^*(BG) \to H^*(BH)$ maps to the invariants $(H^*(BH))^{W_G(H)}$.
When $H$ is abelian, its cohomology is well-known (polynomial tensor exterior) and thus the cohomology of $G$ is mapping to something which can in principal be computed by invariant theory. Follow this with Quillen's theorem that the sum of these maps over all abelian subgroups has kernel which contains only nilpotent elements, and special cases such as Milgram's theorem that this is injective for symmetric groups, and you have a powerful computational tool.