  Let $X$ be a complex smooth quasi-projective variety with a fixed epimorphism
$\nu\colon\pi_1(X)\twoheadrightarrow H$, where $H$ is a finitely generated
abelian group with $\mathrm{rank}H\geq 1$. In this paper, we study the
asymptotic behaviour of Betti numbers with all possible field coefficients and
the order of the torsion subgroup of singular homology associated to $\nu$,
known as the $L^2$-type invariants. When $\nu$ is orbifold effective, we give
explicit formulas of these invariants at degree 1. This generalizes the
authors' previous work for $H\cong \Z$.
