The question is: how does the Frobenius automorphism of $\overline{\mathbb{F}_p}$ act on the prime-to-p $\pi_1(\mathbb{P}^1_{\overline{\mathbb{F}_p}}\setminus\{a_1,...,a_r\})$ where $a_1,...,a_r \in \overline{\mathbb{F}_p}$ (which is isomorphic to the prime-to-p part of the profinite completion of $\langle \alpha_1,...,\alpha_r|\alpha_1...\alpha_r=1\rangle$)?

Of course, I don't actually expect an answer. As I gather from various vague statements in various papers, this is not well understood.

My question is: what is known about it? Where can I read more? And in general any insight about this question is very welcome.

I put a community wiki stamp on this because there's no one right answer.

From Grothendieck's work we know that the prime-to-p fundamental group $\pi_1(\mathbb{P}^1_{\overline{\mathbb{F}_p}}\setminus\{a_1,...,a_r\})$ where $a_1,...,a_r \in \mathbb{F}_p$ is isomorphic to the prime-to-p part of the profinite completion of $\langle \alpha_1,...,\alpha_r|\alpha_1...\alpha_r=1\rangle$.

The question is: how does the Frobenius automorphism of $\overline{\mathbb{F}_p}$ act on the prime-to-p $\pi_1(\mathbb{P}^1_{\overline{\mathbb{F}_p}}\setminus\{a_1,...,a_r\})$ where $a_1,...,a_r \in \mathbb{F}_p$?

I don't actually expect an answer. I gather that this is not well understood.

My question is: what is known about it? Where can I read more? And in general any insight about this question is very welcome.

I put a community wiki stamp on this because there's no one right answer.