Abstract: We study the Bieri-Neumann-Strebel-Renz invariants and we prove the following criterion: for groups H and K of type FPn such that [H, H] subset of K subset of H and a character chi : K -> R with chi([H, H]) = 0, we have [chi] epsilon Sigma(n) (K, Z) if and only if [mu] epsilon Sigma(n) (H, Z) for every character mu : H -> R. that extends chi. The same holds for the homotopical invariants Sigma(n)(-) when K and H are groups of type F-n. We use these criteria to complete the description of the Sigma-invariants of the Bieri-Stallings groups G(m), and more generally to describe the Sigma-invariants of the Bestvina-Brady groups. We also show that the "only if" direction of the above criterion holds if we assume only that K is a subnormal subgroup of H, where both groups are of type FPn. We apply this last result to wreath products Ver menos