Hom spaces over subgroups

Question

I am a bit confused on how to calculate $\operatorname{Hom}_{\mathbb{Z}H}(\mathbb{Z}G,\mathbb{Z})$ for a subgroup $H$ of $G$ where I am considering the integers as the trivial module over $G$. For example if I consider $\mathbb{Z}/4\mathbb{Z}$ generated by an element $\sigma$ and the subgroup isomorphic to $\mathbb{Z}/2\mathbb{Z}=\{1,\sigma^2\}$. As I tried to calculate this, it seems that $\operatorname{Hom}_{\mathbb{Z}H}(\mathbb{Z}G,\mathbb{Z})\cong \mathbb{Z}\oplus\mathbb{Z}$ as the image of $1$ and $\sigma$ are free to go anywhere. I have the feeling that this may be wrong as it doesn't work for calculating the cohomology of the subgroup using the projective resolution with respect to $G$. Any clarifications to my confusion would be very welcome!