What is known about the prime-to-$p$ etale fundamental group of $\mathbb{P}^1_{\mathbb{F}_p}$ minus $\mathbb{F}_p$-rational points?

Is it known to be (the prime-to-$p$ part of the profinite completion of) a finitely presentable group?

Is such a presentation known? Is there a guess for what it is? What is known about it?

• Over separably closed fields, see Cor. 2.12, Exp. XIII, SGA1 (for "tame" $\pi_1$ too, not just prime-to-$p$ $\pi_1$). In general, using 6.1 Exp. IX, one deduces that the tame $\pi_1$ for complement of an etale divisor in a smooth proper geometrically connected curve over a field $k$ is an extension of the Galois group of the ground field $k$ by the tame $\pi_1$ over $k_s$. So this addresses the tame analogue of the question when the structure of ${\rm{Gal}}(k_s/k)$ is well-understood (e.g., pro-cyclic). – nfdc23 Sep 25 '16 at 3:02
• Okay, but is anything known about the action of the Frobenius, or is this shrouded in mystery? Are there any conjectures? Expectations? – Andrew NC Sep 25 '16 at 13:00
• Seems totally mysterious to me, since even the description of the geometric tame $\pi_1$ is very indirect, ultimately via topology over $\mathbf{C}$ which has no visible link with the original outer action by a Frobenius element over $\mathbf{F}_p$. I am not aware of any conjectures or expectations on this, but why not email an anabelian expert? For the "most" that can be seen by algebraic methods alone, see arxiv.org/abs/math/0703139 (which however is over algebraically closed fields, so doesn't help with the mystery about outer action of Frobenius). – nfdc23 Sep 25 '16 at 15:57