I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 130):
Why is $\Pi(n):= \pi_1(\mathbb{P}^1 \backslash \{P_1,..., P_n\})$ a quotient of the profinite Galois group $Gal(\overline{\mathbb{Q}(t)} \vert \mathbb{Q}(t))$?
If $X$ is a Noetherian normal integral scheme with generic point $\eta$ then there is an identification of $\pi_1(X,\overline{\eta})$ with $\mathrm{Gal}(M/k(\eta))$ where $M$ is the compositum of all finite separable extensions $L/k(\eta)$ such that the normalization $Y$ of $X$ in $L$ is a finite etale cover $Y\to X$. In particular, since $M$ is Galois (as can be easily checked) you deduce that $\mathrm{Gal}(\overline{k(\eta)}/k(\eta))$ surjects on to $\pi_1(X,\overline{\eta})$.
In your case note $X=\mathbb{P}^1-\{p_1,\ldots,p_n\}$ is normal and $k(\eta)=\mathbb{Q}(t)$.