On p.64 of the book Cyclotomic Fields and Zeta Values by J. Coates and R. Sujatha: They seemed to have used the argument as follows: Let $G$ be a pro-$p$ group. If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is an exact sequence of $G$-modules, then taking $G$-invariant, we have an exact sequence $C^G\rightarrow A_G\rightarrow B_G$.
I don't know why, could anybody help me? Thanks.