TITLE: Open subgroups of the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$
QUESTION [5 upvotes]: Let $G$ be the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$. Then $G$ is isomorphic to a semidirect product of $\widehat  {\mathbb Z}(1)$ by $ Gal_\mathbb Q$. 
Is it true that every open subgroups of $G$ splits as a semidirect product of  $\widehat  {\mathbb Z}(1)$ by $ Gal_K$ for some finte extension $K$ of $\mathbb Q$?
In general, is there a nice description of the extensions $U$:
$$1\to \widehat  {\mathbb Z}(1)\to U\to Gal_K\to 1?$$

REPLY [2 votes]: The first question seem OK. Here is a sketch. I might have overlooked something.
If $H\subset G$ is an open subgroup, then the image
of $H$ in $Gal_\mathbb{Q}$ is open so necessarily of the form $Gal_K$. So we get an extension
$$1\to H\cap \hat{\mathbb{Z}}\to H\to Gal_K\to 1$$
The kernel $ H\cap \hat{\mathbb{Z}}\subset \hat{\mathbb{Z}}$ is closed and of finite index, so it is   $N\hat{\mathbb{Z}}$ for some integer $N>0$.
Let $H'\subset G$ be the preimage $Gal_K$. The kernel of $H'\to Gal_K$ is $\hat{\mathbb{Z}}$. We have that $H\subset H'$ and $H\cap \ker(H\to Gal_K) = N\hat{\mathbb{Z}}$. We can identify $H'=\pi_1^{et}(\mathbb{G}_{m,K})$. Let $H''\subset H'$ be the subgroup corresponding to the étale cover $\mathbb{G}_{m,K}\to \mathbb{G}_{m,K}$ given by multiplication by $N$.
As a  subgroup $H\subset H'$ coincides with $H''$. Thus
$H\cong \pi_1^{et}(\mathbb{G}_{m,K})$ which a semi direct product of $Gal_K$ with 
$\hat{\mathbb{Z}}(1)$.