Let $X$ be a finite CW-complex of $n$.
For $i\geq 2$, $\pi_i (X)$ is a $\mathbb{Z}\pi_1 (X)$-module.
for $i\geq 2$, $H_i (\tilde{X})$ is a finitely generated $\mathbb{Z}\pi_1 (X)$-module, where $\tilde{X}$ is the universal covering of $X$.
I know that the connection between homotopy groups and homology groups is given by the Hurewicz-Theorem. But my question is that:
My question: Can we express $\pi_i (X)$ (as a $\mathbb{Z}\pi_1 (X)$-module) in terms of $H_i (X)$'s (as $\mathbb{Z}$-module) or $H_i (\tilde{X})$ (as $\mathbb{Z}\pi_1 (X)$-module) for $2\leq i\leq n$? (particularly when they are free).