Put $G= \mathbb{C} \rtimes_{\phi} \mathbb{C} \setminus \{0\}$ where $\phi_{a} (z)= az$ for $a \in \mathbb{C} \setminus \{0\}$. $G$ is a real $4$ dimensional lie group then it has a unique left invariant metric which restrict to the standard Euclidiean metric at the neutral element. Let $I(G)$ be the group of isometries of $G$ with respect to this invariant metric.

On the other hand $G$ is an open subset of the complex Euclidiean space. The group of holomorphic automorphisms of $G$ is denoted by $Aut(G)$.

we constructed $G$ as an obvious generalization of the $2$ dimensional lie group $H= \mathbb{R} \rtimes \mathbb{R} \setminus \{0\}$, the Poincare upper half plane. We know that the group of holomorphic automorphism of $H$ preserve the left invariant metric of $H$.

Now a natural question is that What is the structure of $Aut (G)$?What geometric structure on $G$ is preserved by $Aut(G)$?

Are there any relarion between $I(G)$ and $Aut (G)$?

Is there a presice description for these two groups?