Epimarginal subgroup

Definition
Suppose $$\mathcal{V}$$ is a subvariety of the variety of groups. Consider a group $$G$$ (not necessarily in $$\mathcal{V}$$. The epimarginal subgroup of $$G$$ with respect to $$\mathcal{V}$$ is defined as follows:


 * Consider group extensions $$1 \to A \to E \to G \to 1$$ with the property that the image of $$A$$ in $$E$$ in contained in the $$\mathcal{V}$$-marginal subgroup of $$E$$.
 * For each such extension, consider the image in $$G$$ under the map $$E \to G$$ of the $$\mathcal{V}$$-marginal subgroup of $$E$$.
 * Take the intersection of all possible such images over all possible such extensions.

The intersection thus obtained is the $$\mathcal{V}$$-epimarginal subgroup of $$G$$. Any subgroup that arises as the $$\mathcal{V}$$-epimarginal subgroup for some variety $$\mathcal{V}$$ is termed an epimarginal subgroup.