For a field $k$ of characteristic $0$, there is an exact sequence $$1\to\pi_1(\mathbb A_{\overline k}^1\setminus\{0\})\cong\widehat{\mathbb Z}\to \pi_1(\mathbb A^1_k\setminus\{0\})\to\mathrm{Gal}(\overline k/k)\to 1,$$ so there is an action of $\mathrm{Gal}(\overline k/k)$ on $\widehat{\mathbb Z}$. That is, there is a group homomorphism $\mathrm{Gal}(\overline k/k)\to\widehat{\mathbb Z}^\times$.
In particular, when $k=\mathbb Q$ there is a homomorphism $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\to\widehat{\mathbb Z}^\times$.
Is this the same as the abelianization map $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\to \mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)^{\mathrm{ab}}\cong\widehat{\mathbb Z}^\times$?
Let's think of $\Bbb A^1_\bar{k}\setminus\{0\}$ as $\Bbb G_{m,\bar{k}}$, which is a group scheme.
Every finite étale cover of $\Bbb G_{m,\bar{k}}$ is given by $$m_n:\Bbb G_{m,\bar{k}} \to \Bbb G_{m,\bar{k}}, x \mapsto x^n$$ for some nonzero $n \in \Bbb N$ with automorphism group canonically isomorphic to $\mu_n(\bar{k})$ via the map $$\mu_n(\bar{k}) \to \mathrm{Aut}(m_n), \zeta \mapsto (x \mapsto \zeta x)$$ From this, we see that the action of $\mathrm{Gal}(\bar{k}/k)$ corresponds to the action on $\mu_n(\bar{k})$. Taking the inverse limit yields that the action of $\mathrm{Gal}(\bar{k}/k)$ corresponds to the action on $\mu(\bar{k})$ (i.e. the cyclotomic character), which in the case $k=\Bbb Q$ is also $\mathrm{Gal}(\overline{\Bbb Q}/\Bbb Q) \to \mathrm{Gal}(\overline{\Bbb Q}/\Bbb Q)^{ab} \cong\widehat{\Bbb Z}^\times$.