This question is from Cohomology of Number Fields by Neukirch(page-62).
Let $G$ be a profinite group and $H$ be an arbitrary closed subgroup of $G$. For every discrete $H$-module $A$, define $M= Ind^H_{G}(A)$ consisting of all continuous maps $x:G\rightarrow A$ such that $x(τσ)=τx(σ)$ for all $τ\in H$.Which is a $G$ module and the action of $\rho \in G$ on $M$ is given by $(\rho x)(\sigma)=x(\sigma \rho).$ There a canonical projection $$ Ind^H_{G}(A) \rightarrow A $$ which takes $x \in Ind^H_{G}(A)$ to $x(1) \in A.$ This is a $H$-module homomorphisim.
After that, he is calming that the above map takes the $H$-submodule $A^{'} = \lbrace x:G → A ~|~ x(τ ) = 0 ~\forall τ \not\in H \rbrace$ isomorphically onto $A.$ I don't understand how this is possible. Beacuse if $A^{'}$ contains a nonzero element then H has to a open subgroup.