Let $K$ be a local field, and $\bar{K}$ its algebraic closure. Let $\mathcal{C}$ be the category of (continuous) $G=\text{Gal}(\bar{K}/K)$-modules. Let $M$ be a finite $G$-module. For any injective module $I$ of $\mathcal{C}$,
is $H^r(G, \text{Hom}_{\mathbb{Z}}(M,~I))=0$ for all $r>0$?
(Here on $\text{Hom}_{\mathbb{Z}}(M,~A)$, $g \in G$ acts as $(g\phi)(x)=g(\phi(g^{-1}(x)))$.)