Let $X$ be a connected topological space with abelian fundamental group. Let $\mathcal{L}$ be a $\mathbb{Z}$-valued local system on $X$.

Suppose that I know the full homology $H_*(X;\mathbb{Z})$. Are there any tools which could allow me to compute (some part of) the local-coefficient homology $H_*(X; \mathcal{L})$? For example, if I know that the rank of $H_*(X;\mathbb{Z})$ becomes unbounded as a function of the degree, can I conclude the same about $H_*(X; \mathcal{L})$?