The abelianizer functor is not faithful

Question

I wanted to understand why the abelianizer functor from the category Grp of groups to the category Ab of abelian groups is not faithful. This amounts to finding a couple of groups $G$ and $H$ and two distinct group homomorphisms $f:G\to H$ and $g:G\to H$ that induce the same group homomorphism from $G/[G,G] \to H/[H,H]$. Thanks!

Answer 1

Let $H$ be a perfect group and $G$ be arbitrary, $f,g\colon G\rightarrow H$ be two homomorphisms such that $f$ is trivial and $g$ is nontrivial. Then, the induced maps to abelianizations are both trivial.

Answer 2

For a very simple example, consider the case $G := \mathbb{Z} / 2\mathbb{Z}$, $H := S_3$. Then we can define $f : G\to H$ by $0 + 2\mathbb{Z} \mapsto \operatorname{id}, 1 + 2\mathbb{Z} \mapsto (1,2)$ and $g : G \to H$ by $0 + 2\mathbb{Z} \mapsto \operatorname{id}, 1 + 2\mathbb{Z} \mapsto (1,3)$. Then both $f$ and $g$ induce the identity map on $\mathbb{Z} / 2 \mathbb{Z} \simeq G^{ab} \simeq H^{ab}$.