Homoclinism of groups

Definition
For any group $$G$$, let $$\operatorname{Inn}(G)$$ denote the defining ingredient::inner automorphism group of $$G$$, $$G'$$ denote the defining ingredient::derived subgroup of $$G$$, and $$Z(G)$$ denote the defining ingredient::center of $$G$$.

Let $$\omega_G$$ denote the map from $$\operatorname{Inn}(G) \times \operatorname{Inn}(G)$$ to $$G'$$ defined by first taking the map $$G \times G \to G'$$ given as $$(x,y) \mapsto x^{-1}y^{-1}xy$$ and then observing that the map is constant on the cosets of $$Z(G) \times Z(G)$$.

A homoclinism of groups $$G_1$$ and $$G_2$$ is a pair $$(\zeta,\varphi)$$ where $$\zeta$$ is a homomorphism from $$\operatorname{Inn}(G_1)$$ to $$\operatorname{Inn}(G_2)$$ and $$\varphi$$ is a homomorphism from $$G_1'$$ to $$G_2'$$ such that $$\varphi \circ \omega_{G_1} = \omega_{G_2} \circ (\zeta \times \zeta)$$. In symbols, this means that for any $$x,y \in \operatorname{Inn}(G_1)$$ (possibly equal, possibly distinct), we have:

$$\varphi(\omega_{G_1}(x,y)) = \omega_{G_2}(\zeta(x),\zeta(y))$$

Pictorially, the following diagram must commute:

$$\begin{array}{ccc} \operatorname{Inn}(G_1) \times \operatorname{Inn}(G_1) & \stackrel{\zeta \times \zeta}{\to} & \operatorname{Inn}(G_2) \times \operatorname{Inn}(G_2) \\ \downarrow^{\omega_{G_1}} & & \downarrow^{\omega_{G_2}}\\ G_1' & \stackrel{\varphi}{\to} & G_2'\\ \end{array}$$