  We study the integral, rational, and modular Alexander invariants, as well as
the cohomology jump loci of groups arising as extensions with trivial algebraic
monodromy. Our focus is on extensions of the form $1\to K\to G\to Q\to 1$,
where $Q$ is an abelian group acting trivially on $H_1(K;\mathbb{Z})$, with
suitable modifications in the rational and mod-$p$ settings. We find a tight
relationship between the Alexander invariants, the characteristic varieties,
and the resonance varieties of the groups $K$ and $G$. This leads to an
inequality between the respective Chen ranks, which becomes an equality in
degrees greater than 1 for split extensions.
