Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there is a fundamental exact sequence $$1 \to \pi_1(\overline{X},\overline{x}) \to \pi_1(X,\overline{x}) \to \text{Gal }(\overline{\mathbb{F}_q}/\mathbb{F}_q) \to 1,$$ where we wrote $\overline{X} = X \times \overline{\mathbb{F}_q}$. Passing to abelianizations gives us an exact sequence $$\pi_1(\overline{X},\overline{x})^{\text{ab}} \to \pi_1(X,\overline{x})^{\text{ab}} \to \text{Gal }(\overline{\mathbb{F}_q}/\mathbb{F}_q) \to 1.$$ I don't think this sequence is exact on the left, since by class field theory the kernel of $\pi_1(X,\overline{x})^{\text{ab}} \to \text{Gal }(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ is finite. So what is the kernel of $\pi_1(\overline{X},\overline{x})^{\text{ab}} \to \pi_1(X,\overline{x})^{\text{ab}}$?