Let $K$ be a $p$-adic field with algebraic closure $\overline{K}$. Then if $K^\text{nr}$ is the maximal unramified extension of $K$ in $\overline{K}$, there is an explicit invariant map: $$ H^2(\Gamma_{K^\text{nr}/K}, (K^\text{nr})^{\times}) \to \mathbb{Q}/ \mathbb{Z} $$ given by the composition $$ H^2(\Gamma_{K^\text{nr}/K}, (K^\text{nr})^{\times}) \to H^2(\widehat{\mathbb{Z}}, \mathbb{Z}) \xrightarrow{\delta^{-1}} \operatorname{Hom}(\widehat{\mathbb{Z}}, \mathbb{Q}/\mathbb{Z}) \xrightarrow{\operatorname{ev}(1)} \mathbb{Q}/ \mathbb{Z}. $$
For a finite Galois extension $L/K$ in $\overline K$, I can define an invariant map via the composition $$ H^2(\Gamma_{L/K}, L^{\times}) \hookrightarrow H^2(\Gamma_{\overline{K}/K}, \smash{\overline{K}}^{\times}) \cong H^2(\Gamma_{K^\text{nr}/K}, (K^\text{nr})^{\times}), $$ and then composing with the previously defined map. However it's not so obvious to me on the level of cocycles what this latter map does because the isomorphism at the end is not very explicit.
Does anyone know an explicit way to make sense of this invariant map so that given a cocycle in $H^2(\Gamma_{L/K}, L^{\times})$, one could reasonably compute what class in $\mathbb{Q}/\mathbb{Z}$ it corresponds to?
EDIT: For instance suppose that $L=K(\zeta_{p^n})$. What should the invariant map look like in this case?
