We write $G_w={\rm Gal}(L_w/K_v)$.

> **Definition.**  For $n\ge 1$, we denote
$$Ш_\omega^n(G,M)=\ker\Big(H^n(G,M)\to\prod_C H^n(C,M)\Big)$$
 where $C$ runs over the cyclic subgroups of $G$.
 
> **Remark.** $Ш^2(L/K,M)\subseteq Ш_\omega^2(G, M)$. Indeed,
 by the  Chebotarev density theorem, for any cyclic subgroup $C\subseteq G$
 there exist $v$ and $w$ such that $G_w=C$.
 
>  **Definition.** A permutation $G$-module is a torsion free $G$-module admitting a $G$-invariant (as a subset) ${\mathbb Z}$-basis $B$.
 
> **Theorem.** If $M$ is a permutation $G$-module, then $Ш_\omega^2(G,M)=0$, and therefore $Ш^2(L/K,M)=0$.

*Sketch of proof.* 

  *Step 1.*
 We have
 $$ Ш_\omega^2(G,M_1\oplus M_2)=Ш_\omega^2(G,M_1) \oplusШ_\omega^2(G, M_2),$$
 which reduce the theorem to the case when $G$ acts on the basis $B$ transitively.
 Then using Shapiro's lemma we reduce the theorem to the case $M={\mathbb Z}$ (with trivial $G$-action).
 
 *Step 2.* We have a short exact sequence of $G$-modules (with trivial $G$-action)
 $$0\to {\mathbb Z}\to{\mathbb Q}\to {\mathbb Q}/{\mathbb Z}\to 0,$$
which gives rise to cohomology exact sequences
\begin{align*}
&0= H^1(G,{\mathbb Q})\to H^1(G,{\mathbb Q}/{\mathbb Z}) \to H^2(G,{\mathbb Z})\to H^2(G,{\mathbb Q})=0\\
&0= H^1(C,{\mathbb Q})\to H^1(C,{\mathbb Q}/{\mathbb Z}) \to H^2(C,{\mathbb Z})\to H^2(C,{\mathbb Q})=0.
\end{align*}
Therefore, we may identify
$$Ш_\omega^2(G,{\mathbb Z})=Ш_\omega^1(G,{\mathbb Q}/{\mathbb Z}),$$
and it suffices to show that $Ш_\omega^1(G,{\mathbb Q}/{\mathbb Z})=0$.
By definition, 
$H^1(G,{\mathbb Q}/{\mathbb Z})={\rm Hom}(G,{\mathbb Q}/{\mathbb Z})$ and $H^1(C,{\mathbb Q}/{\mathbb Z})={\rm Hom}(C,{\mathbb Q}/{\mathbb Z})$.
It remain to show that 
$$\ker\Big({\rm Hom}(G,{\mathbb Q}/{\mathbb Z})\to \prod_C{\rm Hom}(C,{\mathbb Q}/{\mathbb Z})\Big)=0,$$
but this is obvious because $G$ is the union of its cyclic subgroups $C$.
 
*Reference:* Lemma 1.9 of J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80.