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 $(g\phi)(x)=g(\phi(g^{-1}(x)))$.)

all$G$-modules? No topology involved, just some abelian group with a $G$-action? $\endgroup$ – user94041 Feb 24 '17 at 16:41