In Iwasawa's paper On Galois groups of local fields, he proves that if $V$ is the maximal tamely ramified extension of $\mathbb{Q}_p$, with Galois group $\Gamma$ over the base, then its abelianized absolute Galois group (i.e., the abelianized wild inertia group of $\mathbb{Q}_p$) sits in a short exact sequence
$$1\to R\to G\to \mathbb{Z}_p(1) \to 1,$$
where $R$ is the regular representation of $\Gamma$ on the set of $p$-adic measures on $\Gamma$, and the conjugation action on $\mathbb{Z}_p(1)$ is given by the usual cyclotomic character and then the identification of $\mathbb{Z}/(p-1)(1)$ with the $(p-1)$st roots of unity on the tame inertia group. (He actually proves a similar thing over any degree $d$ extension of $\mathbb{Q}_p$, where $R$ is replaced by $R^d$.)
Can we explicitly describe the extensions associated to any quotients of this explicitly described group? In particular, I'm asking this question because it feels like what the extension associated to the quotient $\mathbb{Z}_p(1)$ is should be obvious, but it's not. The $p$-cyclotomic extension, after all, should have the trivial conjugation action by cyclotomy.
