Let $G$ be a topological group. In some cases, e.g. when $G$ is discrete or when the spaces $G^n$ are locally contractible and the coefficients are discrete, the cohomology of the classifying space $BG$ is the group cohomology of $G$. So, for simplicity, let us assume that $G$ is discrete.

My question: is there a nice explicit space $B_{\leq k}G$ that is functorial in $G$, such that $H^n(B_{\leq k} G, M) = H^n (BG, M)$ for $n \leq k$ and $0$ for $n>k$? Here $M$ is a $G$-module.

For example, for $G = S^1$, $BG = CP^\infty$ and a possible choice for $B_{\leq 2} S^1$ is $CP^1$. (If it simplifies the question, feel free to assume $M = \mathbf Z$.)

Thank you.

(Edit: In response to Ralph's comment, let us assume $G$ is discrete to simplify, but an answer for non-discrete groups would be interesting, too.)