Let $G$ be a discrete group. Let $H^n(G; \mathbb{Z})$ be the group cohomology of $G$ (with coefficients in $\mathbb{Z}$). Then it's a well-known theorem that $H^n(G; \mathbb{Z})$ is isomorphic to the singular cohomology of the Eilenberg-MacLane $K(G, 1)$
One consequence of this theorem is that the cohomology groups of a $K(G, 1)$ have a purely group-theoretic definition and interpretation. In particular, you can compute the cohomology of $K(G,1)$ simply from information about the group $G.$
My question: do the cohomology groups of $K(G, n)$ for $n>1$ also admit a purely group theoretic definition and/or interpretation?