This is Corollary $9.80$ from Rotman's 'Introduction to Homological Algebra'.
$G$ finite group and $S$ normal subgroup of $G$, $A$ is a $G$-module. Then Hom$_\mathbb{Z}(\mathbb{Z}G, A)^S \cong$ Hom$_\mathbb{Z}(\mathbb{Z}(G/S), A)$. so Hom$_\mathbb{Z}(\mathbb{Z}G, A)$ is $G/S$-coninduced.
In the proof of this, it is stated that it would suffice to show that $(\mathbb{Z}G\otimes_\mathbb{Z}A)^S$ is $G/S$-induced. I don't have a problem with this. However, the author continues to state that each $u\in (\mathbb{Z}G\otimes_\mathbb{Z}A)^S$ has a unique expression $\Sigma_x{_\in}{_G}n_xx$. This last sum I don't quite follow. Not sure why the tensor an element of A didn't appear. Any help would be appreciated!