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 Euclidian space. The group of holomirphic 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?