The \(\Sigma\)-invariants of a group \(H\) are a family of geometric invariants that in particular determine the finiteness properties of all coabelian subgroups of \(H\), that is, subgroups containing the commutator subgroup \([H,H]\). The main result of this paper can be viewed as a generalization of this result to not only understanding the finiteness properties of coabelian subgroups, but also the \(\Sigma\)-invariants of many of the coabelian subgroups themselves. More precisely, given \([H,H]\le K\le H\) with \(H\) of type \(FP_n\) (resp. \(F_n\)), it is a classical result that \(K\) is of type \(FP_n\) (resp. \(F_n\)) if and only if \([\chi]\in\Sigma^n(H,\mathbb{Z})\) (resp. \([\chi]\in\Sigma^n(H)\)) for all characters \(\chi\colon H\to \mathbb{R}\) with \(\chi(K)=0\). (See the paper for definitions and terminology.) The main result here (Theorem 1.1) is that if \(K\) is of type \(FP_n\) (resp. \(F_n\)) and \(\chi\colon K\to\mathbb{R}\) satisfies \(\chi([H,H])=0\), then \([\chi]\in\Sigma^n(K,\mathbb{Z})\) (resp. \([\chi]\in\Sigma^n(K)\)) if and only if \([\mu]\in\Sigma^n(H,\mathbb{Z})\) (resp. \([\mu]\in\Sigma^n(H)\)) for all characters \(\mu\colon H\to\mathbb{R}\) extending \(\chi\). The main application of this result is a computation of the \(\Sigma\)-invariants of Bestvina–Brady subgroups of right-angled Artin groups. Another important, related result (Theorem 1.5) is that if the restriction of a character of a type \(FP_n\) group to a subnormal subgroup of type \(FP_n\) is in the homological \(\Sigma^n\)-invariant of that subgroup, then the original character is in the homological \(\Sigma^n\)-invariant of the group. This has nice applications to permutational wreath products (Corollary 1.6).