This is not an answer but only a too long comment.

While I dont know the answer for this question, I would split it up into two parts: one containing group theory and one containing representation theory.

Given another group $H$, we can assign to any group $G$ the natural map $$ \prod_{f\in Mor(G,H)} f:G\to \prod_{Mor(G,H)} H$$ and look at its image (denoted by $G_H$). Thinking of this image as $G/\bigcap_{f\in Mor(G,H)}Ker(f)$, we see that $G\to G_H$ is natural in $G$.

Has this already been studied. For example I bet someone already looked at the case of $G=F_3$ and $H=F_2$?

In the example where $G$ is finite and $H=k^*$ where $k$ is an algebraically closed field of characteristic zero, $H$ is so large that $G_H$ is naturally isomorphic to $G_{ab}$. This works for any abelian group containing $\mathbb{Z}$ and all $\mathbb{Z}/p$'s; it is nothing special about algebraically closedness and characteristic zero.

Can representation theory be used to understand the case of $H=GL_n(k)$ for $k\ge 0$?