Let $G$ be a finite group.

$$
\Bbb{Z}[G] \xrightarrow{\pi} G^{\text{ab}}, \\
\pi(\sum_{g \in G} x_g g) = \prod_{g \in G} \tilde{g}^{2x_g},
$$
is a group hom, where $\tilde{g} = $ the coset of $g$ in $G$'s abelianization, $G^{\text{ab}}$. $\ker \pi = \{ x \in \Bbb{Z}[G] \ | \ \prod \tilde{g}^{2x_g} = \tilde{1}\}$, but,

$$
\tilde{g}_1^{2 x_1} \cdots \tilde{g}_n^{2 x_n} = \tilde{1} \iff \\
\tilde{g}_1^{x_1} \cdots \tilde{g_n}^{x_n} = \tilde{g_1}^{-x_1} \cdots \tilde{g_n}^{-x_n} \iff \prod_{i = 1..|G|} \tilde{g_i}^{x_i} = \text{ its inverse}