Background:

I've often heard that the fundamental group of $\mathbb{P}^1/\mathbb{Q}-\{0,1,\infty\}$ is extremely hard to understand. First of all, it has a surjective map to the galois group $\rm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, and then you further have a map $$\rm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\rm{Out}(\widehat{\rm{F}_2})$$ to try and get at.

I'm trying to look at the simpler case of working over $\mathbb{F}_p$ instead of $\mathbb{Q}$. Of course, this acquires problems of its own, as degree $p$ extensions in char $p$ are hard. In fact, I'm aware that Tamagawa and many others have results that say you can use the absolute galois group of curves (possibly other varieties too) in char. $p$ to recover the curve, and they heavily use degree $p$ extensions. For this reason, I'm restricting my question to the prime-to-$p$ part of the fundamental group.

Question:

What is the prime-to-$p$ part of $\pi_1\left(\mathbb{P}^1/\mathbb{F}_p-\{0,1,\infty\}\right)$?

If, as I suspect, the above is too difficult, I would be interested in any theorems pertaining to the structure of this group. If, however, this question turns out to be doable, I am equally interested in the similar questions with more points removed from $\mathbb{P}^1$.

Thank you!

For reference: In case people are unfamiliar, the prime-to-$p$ part of a profinite group $G$ is the inverse limit over all finite groups $H$ that have order prime to $p$ and which are continuous quotients of $G$.