the notations follow Serre's famous book Galois cohomology. Suppose $H$ is an open subgroup of a profinite group $G$. Denote by $M^H _G (\mathbb{Z})$ the induced module of the trivial $H$-module $\mathbb{Z}$. Let $A$ be a discrete $G$-module. The exercise is: prove $M^H _G (\mathbb{Z})\otimes_\mathbb{Z} A \simeq M^H _G (A)$ as $G$-modules.
So I find the induced module $M^H _G (\mathbb{Z})$ is a free Abelian group because each element in it is constant on the (right) cosets of $H$. Choose a set of representatives $x_i, i=1,...,n$. Then as an Abelian group $M^H _G (\mathbb{Z})$ has a dual basis $\phi_i, \phi_i (h x_i)=\delta_{ij}, \forall h\in H$.
Now I have to construct $f: M^H _G (\mathbb{Z})\otimes_\mathbb{Z} A \longrightarrow M^H _G (A)$, so this means to determine $f(\phi_i \otimes a)$. But $A$ isn't necessarily a discrete $G$-module, I have no idea how to define this, can anyone give me some help or hint? thanks!