I am reading Milne's notes on Class Field Theory. https://www.jmilne.org/math/CourseNotes/CFT.pdf
On page 99, in the section 'The Local Artin Maps' he says,
Let $L$ be a finite unramified extension of a local field $K$ with Galois group $G$. Then the pair $(G, L^{\times})$ satisfies the hypothesis for Tate's theorem.
Tate's theorem requires $G$ to be finite which it is and also for each $H \leq G $ we want $H^1(H, L^{\times})=0 $ and $H^2(H, L^{\times})$ is cyclic and of order $|H|$.
Tate's theorem then gives us that $$H^r_T(G, \mathbb Z) \cong H^{r+2}_T(G,L^{\times}) $$
I am trying to understand why this holds i.e. why Tate's theorem's hypothesis is being satisfied by $(G, L^{\times})$.
We know $H^1(G,L^{\times})=0 $ from Hilbert's theorem 90. But how to prove this for subgroups $H$ of $G$? It is possible to deduce if we had a way to extend homomorphism form $H$ to $G$.
I have not got $H^2$ either.
Feel free to give any reference. Any help is appreciated.