I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes_n\mathbb{Z}$, say $$\mathbb{Z}^2\rtimes_n\mathbb{Z}=\langle x,y,z: xz=zx, zy=yz, xy=yz^nx\rangle$$ for some natural number $n>0$.

For example, $G$ could be the discrete Heisenberg group, i.e., $$G=\langle x,y,z: xz=zx, zy=yz, xy=yzx\rangle$$

Then it is clear that we can extend linearly an automorphism $\phi$ of $G$ to an automorphism of $\mathbb{Z}G$, we can also define an automorphism by sending $ x$ to $\pm x$, $y$ to $\pm y$, and fixing $z$, but are there any other nontrivial automorphisms?

How to find $\operatorname{Aut}(\mathbb{Z}G)$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$?