Pseudoverbal subgroup

Definition
Suppose $$\mathcal{V}$$ is a subpseudovariety of the variety of groups, i.e., $$\mathcal{V}$$ is a collection of groups closed under taking subgroups, quotients, and direct products. Equivalently, the group property of being in $$\mathcal{V}$$ is a pseudovarietal group property.

The $$\mathcal{V}$$-pseudoverbal subgroup of a group $$G$$ is defined as the intersection of all normal subgroups $$N$$ of $$G$$ for which the quotient group $$G/N$$ is in $$\mathcal{V}$$. Note that the quotient group of $$G$$ by its $$\mathcal{V}$$-pseudoverbal subgroup need not itself be in the pseudovariety.