Let $ L/K $ be a Galois extension of number fields. Let $ G = \mathrm{Gal}(L^{\text{ab}}/K)$, $ H = \mathrm{Gal}(L^{\text{ab}}/L) $ where $ L^{ \text{ab}} $ is the maximal abelian extension of $L $. Then, the conjugation action of $ G $ on $ H $ factors through $ G / H = \mathrm{Gal}(L/K) $, since $ H $ is abelian.
Question: Can we describe an action of $ G/H $ on $ \mathbb{I}_{L} $ (or on $ \mathbb{I}_{L}/L^{\times}$), the idele class group of $ L $ so that the Artin map $$ \mathrm{Art}{_{L/K}} : \mathbb{I}_{L}/L^{\times} \to H $$ is $ G/H $ equivariant?
As pointed out by @franz lemmermeyer, this is actually the job of the Shafarevich-Weil theorem. Curiously, S-W does not seem to be widely known, although it is an important feature of the so called theory of Weil groups (Artin-Tate, chapter 14), which"contains the entire theory of the reciprocity law,[whose results] are wrapped up in one neat abelian bundle, namely a suitable Weil group"(p. 246). Let me try to summarize the content of the S-W theorem.
Consider extensions $M/L/K$ of number fields, where $L/K$ is Galois with group $G$, $M/K$ is Galois with group $E$ and $M$ is a class field over $L$ with abelian Galois group $A\cong C_L/N_{M/L} C_M$, where $C_{*}$ denotes the idèle class group. In the theory of group extensions with abelian kernels (see e.g. A-T, chapter 13), the equivalence class of an exact sequence (*) $1 \to A \to E \to G \to 1$ with abelian $A$ is characterized by a cohomology class $\epsilon \in H^2 (G,A)$. In our arithmetic situation here, a central cohomological result in CFT states that $H^2 (G,C_L)$ is cyclic of order $[L:K]$, with a canonical generator $u_{L/K}$ (usually called the "invariant" of $L/K$). Then the S-W theorem asserts that $\epsilon = \Psi (u_{L/K})$, where $\Psi: H^2 (G,C_L) \to H^2 (G,A)$ is the cohomological map induced by the Artin map $ C_L \to A$. For all the technical details, see e.g. Cassels-Fröhlich, chapter 7, thm. 11.5. See also the interesting historical developments on the Weil groups on p.200.