Let $E/L/K$ a tower of finite abelian extensions of local fields, $G = \operatorname{Gal}(E/K), H = \operatorname{Gal}(E/L)$. There is of course a natural quotient in Tate cohomology: $$ \phi_\star: \widehat{H}^0(G, E^\star) = K^\star/N_{E/K} E^\star \to K^\star/N_{L/K} L^\star = \widehat{H}^0(G/H, L^\star). $$ Question 1: Does this map have any natural group-cohomological interpretation? Does it extend to a morphism of $\widehat{H}^q$ for $q \neq 0$?
This map is not any of the standard change of group maps: restriction, corestriction, inflation (goes the wrong way) or coinflation (only exists for homology) and moreover inflation and coinflation don't descend to Tate cohomology. The essential property needed to define it is compatibility of the norm in towers, which the definitions of modified cohomology don't seem to capture nicely. Is this map actually a natural thing to consider?
My motivation for asking is the following application to local class field theory: one wishes to prove compatibility of the local reciprocity map in towers: $$ \require{AMScd}\begin{CD} K^\star/N_{E/K} E^\star @>{\omega_{E/K}}>> G;\\ @VVV @VVV \\ K^\star/N_{L/K}L^\star @>{\omega_{L/K}}>> G/H; \end{CD} $$ If the reciprocity map is defined using the Tate-Nakayama theorem as a cup-product, then this is not obvious, but is taken as a corollary of the character formula \begin{equation}\tag{$\star$} \chi(\omega_{L/K}(a)) = \operatorname{inv}_K(a \cup d(\chi)) \end{equation} (as in Serre, Local Fields, XI,§3, after Prop. 2), where $\chi \in H^1(G,\mathbb{Q}/\mathbb{Z})$ is a character and $d : H^1(G, \mathbb{Q}/\mathbb{Z}) \to H^2(G, \mathbb{Z})$ is the boundary isomorphism.
Question 2: How does $(\star)$ imply compatibility in towers?
The intended process seems to be as follows: if we define $$\phi^\star : H^1(G/H, \mathbb{Q}/\mathbb{Z}) \to H^1(G, \mathbb{Q}/\mathbb{Z}) $$ to be the pullback of characters (which should surely be inflation), then it suffices to show a projection formula relating $\phi_\star$ and $\phi^\star$: for every $a \in \widehat{H}^0(G, E^\star)$ and $\chi \in H^1(G/H, \mathbb{Q}/\mathbb{Z})$ we would like $$a \cup d(\phi^\star(\chi)) = \operatorname{Inf}(\phi_\star(a) \cup d(\chi))$$ inside $H^2(G, E^\star) = \operatorname{Br}(E/K)$.
Question 3: Does such a formula hold? Why? There is such a formula for restriction and corestriction, but $\phi_\star$ is a strange map from the perspective of cohomology, so I don't necessarily expect it to behave with cup products.