Let $F/\mathbb Q_p$ be a finite extension, and let $I_F=\operatorname{Gal}(\overline F/F^{\mathrm{unr}})\subset\operatorname{Gal}(\overline F/F)$ be the inertia subgroup.
Is there a description of the abelianization $I_F^{\mathrm{ab}}$ known?
Letting $F^{\mathrm{tame}}$ be the maximal tamely ramified extension of $F$, we have an abelian extension $F^{\mathrm{tame}}(\mu_{p^\infty})/F^{\mathrm{unr}}$. How much larger is the full maximal abelian extension of $F^\mathrm{unr}$? Also note that a description of the abelianization of the wild inertia $P_F$ is known due to Iwasawa, see Explicit construction of abelian wild inertial extensions of maximal tamely ramified extension of $\mathbb{Q}_p$?.
