Let $C/\mathbb Q$ be a smooth projective curve of genus $g\geq 2$ or a smooth affine curve of genus $g \geq 1$. The exact sequence

$1 \to \pi_1^{et}(C \otimes_\mathbb Q \bar{\mathbb Q}) \to \pi_1^{et}(C) \to \operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q) \to 1$

gives a homomorphism from $\operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q)$ to the outer automorphism group of $\pi_1^{et}(C \otimes_\mathbb Q \bar{\mathbb Q})$.

Is this homomorphism always injective?

If one instead takes $C$ to be a curve of genus $0$ with $3$ points removed, it is injective, by Belyi's Theorem.