I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where $G$ is a profinite group and $A$ is a $G$ module.
By definition $C^n(G,A)=X^n(G,A)^G$. Here $X^n(G,A)$ is defined to be continuous map from $G^{n+1}$ to $A$ with discrete topology on $A$. Now from page $32$ of the same book using proposition $1.3.7$, I deduce that $X^n(G,A)=X^n(G,Ind_G(A))^G$ that is I get that $C^n(G,A)=({X^n(G,Ind_G(A))^G})^G$. But $(-)^G$ is left exact and I know that $A \rightarrow Ind_G(A)$ is exact. I am lost after that. Any help will be very favorable.