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?



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> How to find $\operatorname{Aut}(\mathbb{Z}G)$ for $G=\mathbb{Z}^2\rtimes\_n\mathbb{Z}$?
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$Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$
Let $G\_n$ the group generated by $x$, $y$, $z$ with $z$ central and $yx=xyz^n$; this is your group up to renaming; I like this one better :-) 


The only finite conjugacy classes of the group $G\_n$ are those of the powers of $z$. It follows that the center of the group algebra $\def\ZZ{\mathbb Z}\ZZ G$ is $\ZZ Z$ with $Z=(z)\subseteq G\_n$ the center. An automorphism $\alpha$ of $\ZZ G\_n$ must preserve the center of the ring, so it induces an automorphism of $\ZZ Z$, and the restriction of $\alpha$ to $\ZZ Z$ must in turn restrict to the group of units of $\ZZ Z$, which is $\pm Z$. The group structure of $\pm Z$ is very simple, and allows us to see that $\alpha$ must map $z$ to $z$ or to $-z$.


*Let me assume now that $n$ is even.* In this case, the ring $\ZZ G\_n$ has an automorphism which maps $x$, $y$ and $z$ to $-x$, $-y$ and $-z$. Composing $\alpha$ with it, we can suppose that in fact $\alpha(z)=z$.


In that case, $\alpha$ preserves the ideal $I\subset\ZZ G\_n$ generated by $z-1$, so passes on to the quotient $\ZZ G\_n/I$, which is isomorphic to the group algebra of the abelian free group generated by the images of $x$ and $y$. The automorphisms of this quotient are easily described, I guess, so one can lift this information to $\ZZ G\_n$. I'll try to check/fill in the details later.


