I am trying to recover the Galois group of the extension $E/F$, where $E$ and $F$ are the fields defined below.
$F$ is a finite extension of $\mathbb{Q}_p$, containing a primitive root of unity $\zeta_p$ and we let $\pi$ be a fixed uniformizer of $F$, (assume $p$ is a prime greater than 2).
$E = \bigcup_{n\geqslant 1}F(\zeta_{p^n}, \sqrt[p^n]{\pi})$, such that $(\zeta_{p^{n+1}})^p = \zeta_{p^n}$ and $(\pi_{n+1})^p=\pi_n$ for all $n\geqslant 1$.
My thinking is to consider the subfield $F_{\infty}= \bigcup_{n\geqslant 1}F(\zeta_{p^n})$, then $F_{\infty}/F$ is the cyclotomic extension of $F$, its Galois group is isomorphic to an open subgroup of $\mathbb{Z}_p^*$, and we can fix a topological generator $\gamma$ of $\operatorname{Gal}(F_{\infty}/F)$ such that for all $n\geqslant 1$ $\gamma(\zeta_{p^n})=\zeta_{p^n}^x$ for some $x\in (\mathbb{Z}/p^n\mathbb{Z})^*$.
I can then consider the extension $E/F_{\infty}$, this extension is the compositum of finite Kummer extensions and so we can fix a topological generator $\sigma$ of $\operatorname{Gal}(E/F_{\infty})$ such that $\sigma(\pi_n) = \zeta_{p^n}\pi_n$ for all $n\geqslant 1$.
The Galois correspondence then gives us an exact sequence
$1 \longrightarrow \langle \sigma \rangle \longrightarrow Gal(E/F) \longrightarrow \langle \gamma \rangle \longrightarrow 1$
This is as far as I get! It seems my next step should be to consider a lift $\tilde{\gamma}$ of $\gamma$, we can define such a lift by stating its action on the elements $\pi_n\in E$. It also seems sensible that upon defining a lift $\tilde{\gamma}$ to consider if the exact sequence splits.
Perhaps I am missing something simple, but I have become quite lost on this any any pointers would be gratefully received.
EDIT: I have come across a paper that considers the same extension "An explicit formula for the Hilbert symbol of a formal group" - Floric Tavares Ribiero. https://hal.inria.fr/file/index/docid/379787/filename/explicit_formula_Hilbert_symbol_formal_group.pdf
On page 10 (but using the notation as above) the group is given as as a semidirect product $\mathbb{Z}_p \rtimes \operatorname{Gal}(F_{\infty}/F)$. The author considers the absolute Galois group $G_F$ of the field $F$ from above. They state that the cyclotomic character $\chi : G_F \rightarrow \mathbb{Z}_p^*$ factorises through $\operatorname{Gal}(E/F)$, and the map $\varphi:G_K \rightarrow \mathbb{Z}_p$ defined for all $g \in G_F$ by $g(\pi_{p^n})= \pi_{p^n}\zeta_{p^n}^{\varphi(g)}$ also factorises through $\operatorname{Gal}(E/F)$. I don't see necessarily though that the group is a semi direct product.
If the main point consists in lifting a topological generator of Gal(F_infty /F), it's very easy because this group is isomorphic to ($Z_{p}$, +), which is free in the category of pro-p-groups. Of course a lift which is more seriously related to the data would be needed to deal with more specific problems, but as far as the semi direct structure alone is concerned...