Isologism of groups

Definition in terms of a defining set of words
Consider a subvariety $$\mathcal{V}$$ of the variety of groups. Denote by $$W$$ a set of words that generate the variety $$\mathcal{V}$$ (i.e., a group is in $$\mathcal{V}$$ iff all words from $$W$$ are trivial for all tuples of elements from the group).

Consider any group $$G$$ (not necessarily in $$\mathcal{V}$$). Denote by $$V^*(G)$$ the defining ingredient::marginal subgroup of $$G$$ with respect to the variety $$\mathcal{V}$$ and denote by $$V(G)$$ the defining ingredient::verbal subgroup of $$G$$ with respect to $$\mathcal{V}$$.

For every word $$w \in W$$, let $$n_w$$ be the number of distinct letters used in the word. $$w$$ defines a $$n_w$$-ary set map:

$$\beta_w: G^{n_w} \to G$$

By the definitions of marginal and verbal subgroup, the map descends to a set map:

$$\gamma_w: (G/V^*(G))^{n_w} \to V(G)$$

An isologism of groups $$G$$ and $$H$$ with respect to $$\mathcal{V}$$ is a pair $$(\zeta,\phi)$$ where $$\zeta$$ is an isomorphism between $$G/V^*(G)$$ and $$H/V^*(H)$$, $$\phi$$ is an isomorphism between $$V(G)$$ and $$V(H)$$, and for every $$w \in W$$, we have:

$$\gamma_w(\zeta(x_1), \zeta(x_2), \dots, \zeta(x_{n_w}) = \phi(\gamma_w(x_1,x_2,\dots, x_{n_w})) \ \forall \ (x_1,x_2,\dots,x_n) \in (G/V^*(G))^{n_w}$$

Two groups are termed isologic groups with respect to $$\mathcal{V}$$ if there exists an isologism with respect to $$\mathcal{V}$$ between them.

Note that the choice of $$W$$ does not matter for this definition, all that matters is that $$W$$ generate the variety $$\mathcal{V}$$.

Definition in terms of homologism
An isologism is an invertible defining ingredient::homologism, i.e., a homologism where both the component homomorphisms are isomorphisms.

Facts

 * Isologism with respect to variety is isologism with respect to any bigger variety