Isologic groups

Definition
Let $$\mathcal{V}$$ be a subvariety of the variety of groups. Two groups $$G$$ and $$H$$ (not necessarily in $$\mathcal{V}$$) are termed isologic with respect to $$\mathcal{V}$$ if there exists an defining ingredient::isologism between them with respect to $$\mathcal{V}$$.

Being isologic can roughly be thought as being congruent modulo $$\mathcal{V}$$, i.e., the difference between the groups lives in $$\mathcal{V}$$. In this sense, it behaves like congruence mod n.

Facts

 * Any group in $$\mathcal{V}$$ is isologic to the trivial group.
 * Isologism with respect to variety is isologism with respect to any bigger variety
 * Isologic groups with respect to fixed nilpotency class lower than theirs have equal nilpotency class