I want to make sure that I am understanding correctly the idea of a homomorphism $(H \leq G) \xrightarrow{\quad \varphi \quad} X$ being "extended" to a homomorphism $G \xrightarrow{\quad \psi \quad} X$. I'll first take an inclusion arrow $H \xrightarrow{\quad i \quad} G$. Then I'll take elements $h \in H \cap G$ and $g \in (G - H \cap G) $. Then $\psi$ is an "extension" of $\varphi$ which changes the domain $H$ to a group $G$ that contains $H$, such that $\psi(h) = \psi(i(h)) = \varphi(h)$. In addition though, we can also perform an operation $\psi(g)$ that cannot be done with $\varphi$. I've attempted to illustrate this with the diagram below. Let a black path commute with any blue or red path, but do not allow any path that has a blue arrow to commute with any path that has a red arrow. Then the extension I outlined above can be represented as such:
Does this correctly represent the principle of extension of a homomorphism?