Let $X$ smooth curve over a finite field $\mathbb{F}_q$ of type $(g,n)$ - that is, $X$ is an open subscheme of its genus $g$ compactification obtained by removing $n$ points.

Any such curve determines an exact sequence of etale fundamental groups

$$1\rightarrow \pi_1(X_{\overline{\mathbb{F}}_q})\rightarrow \pi_1(X)\rightarrow\underbrace{\text{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}_{G_{\overline{\mathbb{F}}_q}}\rightarrow 1$$ whence a representation $$\rho' : G_{\overline{\mathbb{F}}_q}\rightarrow\text{Out}(\pi_1(X_{\overline{\mathbb{F}}_q}))\rightarrow \text{Out}(\pi_1'(X_{\overline{\mathbb{F}}_q}))$$ where $\pi'$ denotes the prime-to-$p$ fundamental group. (Actually since the absolute galois group of a finite field is projective, the exact sequence is split, and so we get an actual representation into $\text{Aut}(\cdots)$)

Are there any examples where $X$ is hyperbolic (equivalently, $\pi_1'(X_{\overline{\mathbb{F}}_q})$ is nonabelian) such that the representation $\rho'$ is known?

This is equivalent to understanding the action of Frobenius on $\pi_1'(X_{\overline{\mathbb{F}}_q})$.

References would also be appreciated.

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