Torsion cohomology for solvable groups of finite rank

We define a class $\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\mathcal{U}$ and $A$ a $\mathbb ZG$-module. If $A$ is $\mathbb Z$-torsion-free and has finite $\mathbb Z$-rank, we stipulate a condition on $A$ that guarantees that $H^n(G,A)$ and $H_n(G,A)$ must be finite for $n\geq 0$. Moreover, if the underlying abelian group of $A$ is a \v{C}ernikov group, we identify a similar condition on $A$ that ensures that $H^n(G,A)$ must be a \v{C}ernikov group for all $n\geq 0$.