Homologism 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:

$$\beta_{w,G}: G^{n_w} \to G$$

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

$$\gamma_{w,G}: (G/V^*(G))^{n_w} \to V(G)$$

A homologism of groups $$G_1$$ and $$G_2$$ with respect to $$\mathcal{V}$$ is a pair $$(\zeta,\varphi)$$ where $$\zeta: G_1/V^*(G_1) \to G_2/V^*(G_2)$$, $$\varphi: V(G_1) \to V(G_2)$$ are homomorphisms, and for every $$w \in W$$, we have:

$$\gamma_w(\zeta(x_1), \zeta(x_2), \dots, \zeta(x_{n_w}) = \varphi(\gamma_w(x_1,x_2,\dots, x_{n_w})) \ \forall \ (x_1,x_2,\dots,x_n) \in (G_1/V^*(G_1))^{n_w}$$

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

In other words, for every $$w \in W$$, the following diagram commutes:

$$\begin{array}{ccc} (G_1/V^*(G_1))^{n_w} & \stackrel{\zeta \times \zeta}{\to} & (G_2/V^*(G_2))^{n_w} \\ \downarrow^{\gamma_{w,G_1}} & & \downarrow^{\gamma_{w,G_2}}\\ V(G_1) & \stackrel{\varphi}{\to} & V(G_2)\\ \end{array}$$

Note that the choice of the defining set of words does not matter, i.e., if $$W_1$$ and $$W_2$$ are different sets of words that generate the same variety $$\mathcal{V}$$, the condition of being a homologism with respect to $$W_1$$ coincides with the condition of being a homologism with respect to $$W_2$$.