I am trying to study the representation of the Weil group from the book "The Local Langlands Conjecture for $GL(2)$". I have some problem with the topology of this group.

Let $F$ be a non archimidian local field, so a finite extension of $\mathbb{Q}_p $ for a prime $p$. If $\overline{F}$ is an algebraic closure of $F$ and $F_{\infty}$ is the maximal unramified extension of $F$ in $\overline{F}$, we have an exact sequence $$1 \to \mathcal{Gal}(\overline{F}/F_{\infty})\to \mathcal{Gal}(\overline{F}/F)\to \mathcal{Gal}(F_{\infty}/F) \to 1.$$

Since ${\mathcal{Gal}(\overline{F}/F})\cong \lim \mathbb{Z}/m\mathbb{Z}$ and in $\mathbb{Z}/m\mathbb{Z}$ we have the Frobenius element, we have an element $\Phi_F=(\Phi_m)_m\in \lim \mathbb{Z}/m\mathbb{Z}$, called geometric profenious substitution where $\Phi_m$ is the inverse of the Frobenius element in $\mathbb{Z}/m\mathbb{Z}$. We define the Weil group as the topological group that satisfies the exact sequence $$1 \to I_F \to W_F \to <\Phi_F> \to 1$$ with $I_F=\mathcal{Gal}(\overline{F}/F_{\infty})$. The topology of $W_F$ is the topology in which $I_F$ is open, $I_F$ in $W_F$ has the same topology of $I_F$ in $\mathcal{Gal}(\overline{F}/F)$ and the map $\iota:W_F \to \mathcal{Gal}(\overline{F}/F)$ is continuous.

I am trying to understand who are the open subgroup of $W_F$. Of course, since $\iota:W_F \to \mathcal{Gal}(\overline{F}/F)$ is a continuous map, all the open subgroups of $\mathcal{Gal}(\overline{F}/F)$ are open subgroups of $W_F$. So all the groups of the form $W_F \cap \mathcal{Gal}(\overline{F}/E)$ with $E/F$ finite extension are open subgroups of $W_F$. In the book there is a (not proved) proposition that classify the open subgroup of finite index of $W_F$: they are of the form $W_F \cap \mathcal{Gal}(\overline{E}/E)$ for some finite extension $E$ of $F$, so the ones that I declared before.

Of course we have other open subgroup, infact $I_F$ has to be open (it is also closed) since $W_F/I_F$ is discrete, but in $\mathcal{Gal}(\overline{F}/F)$ it is closed but not open.

I would a better classification of the open subgroup of $W_F$ and their connection with $I_F$. For example, in the book that I mentioned in the first part of my question, when it takes a smooth representation $(V,\pi)$ of $W_F$, it assume that the open subgroup that fix an element $v \in V$ is of the form $I_F \cap \mathcal{Gal}(\overline{E}/E)$ for some finite extension $E/F$, and not only an open compact subgroup of $W_F$.

I am sure that there is a deep connection with the fact that $$W_F=<I_F,\sigma>$$ where $\sigma$ is a lift of $\Phi$. But it is not clear to me the connection between the open of $W_F$ and $I_F$.