Let $K$ be a local field. Let $K^{nr}$ and $K^t$ be its maximal unramified and tamely ramified extensions, respectively. One can show that $\operatorname{Gal}(K^{nr}/K) \cong \widehat{\mathbb Z}$ and that $\operatorname{Gal}(K^{nr}/K)$ is topologically generated by the Frobenius element.
Theorem: $\operatorname{Gal}(K^t/K^{nr}) \cong \prod_{\ell \neq p} \mathbb Z_\ell$ and $\operatorname{Gal}(K^t/K) \cong \prod_{\ell \neq p} \mathbb Z_\ell \rtimes \widehat{\mathbb Z}$, where the action of $\widehat{\mathbb Z}$ is determined by the Frobenius acting by conjugation.
Kevin Buzzard cites this theorem in his lecture series on the Langlands programme (available on youtube). It should follow fairly easily from the Sylow theorems for profinite groups.
However I have not been able to find a reference for it in any textbooks! Can somebody please help me out?
Edit: The theorem is covered in Neukirch-Schmidt-Wingberg's "Cohomology of Number Fields" as Proposition 7.5.2. Credit goes to Tom Fisher.
Best Answer
As usual let $K$ be a finite extension of $\Bbb{Q}_p$. Then $$K^{nr} =\bigcup_{p\,\nmid \,n} K(\zeta_n), \qquad K^t =\bigcup_{p\,\nmid \,n} K(\pi_K^{1/n})=\bigcup_{p\,\nmid \,n} K(p^{1/n})$$
$$Gal(K^{t}/K^{nr}) = \{ (\pi_K^{1/n} \to \zeta_n^{a\bmod n} \pi_K^{1/n}), a\in \widehat{\Bbb{Z}}/\Bbb{Z}_p\}$$
$Gal(K^{nr}/K)$ is a finite index subgroup of $\widehat{\Bbb{Z}}^\times/\Bbb{Z}_p^\times$, choosing the natural Frobenius of $K^t$ gives an embedding $Gal(K^{nr}/K)\to Gal(K^{t}/K)$ from which you get your $\rtimes$.