Let $L/K$ be an extension of number fields, $v$ be a place of $K$ and $T$ a $G_K$-module. Shapiro's lemma yields isomorphisms
$H^1(K,Ind_L^K T)\longrightarrow H^1(L,T)$,
$H^1(K_v, Ind_L^K T)\longrightarrow \bigoplus_{w\mid v} H^1(L_w,T)$
for $w$ varying over all places of $L$ above $v$ (this last isomorphism comes e.g. from [Rubin, Euler systems, Proposition 4.2]).
One could close the above diagram with two natural vertical restriction (or, localization, if you prefer) maps in cohomology. Is that diagram commutative?
More generally, is it true that there is a similar commutativity with any restriction or inflation map in group cohomology?