Let $R=\Bbb{Z}[C_2]\cong \Bbb{Z}[x]/(x^2-1)$, and consider $\Bbb C^\times$ as an $R$-module by letting $x$ act by conjugation. I would like to determine the generators of the $R$-module $$H:=\hom_{R}(\Bbb{Z}[y,z]/(y^2-1,z^2-1),\Bbb C^\times),$$ where $x$ acts on $S=\Bbb{Z}[y,z]/(y^2-1,z^2-1)$ by multiplying by $yz$. Note that this means that $S$ is generated as an $R$-module by $1$ and $y$, where $x\cdot 1= yz$ and $x\cdot y= z$.
Then, an $R$-module homomorphism $S\to \Bbb C^\times$ is determined by where we send $1$ and $y$. I then label an arbitrary such morphism by $f_{c_1,c_2}$ where $c_1,c_2\in\Bbb C^\times$ and $f_{c_1,c_2}(1) =c_1$ and $f_{c_1,c_2}(y)=c_2$.
Then $(f_{c_1,c_2}+f_{d_1,d_2})=f_{c_1d_1,c_2d_2}$ and $x\cdot f_{c_1,c_2}=f_{\overline{c_1},\overline{c_2}}$. As a set $H$ should be in bijection with $(\Bbb C^\times)^2$, but I can't see what the generators are as an $R$-module.