Every Lie group $G$ is contained in its holomorph, Hol($G$) = $G \rtimes $ Aut($G$), in a natural way.

Is Hol$(G)$ always a Lie group?

If the answer is yes our main questions:

1.Is the left invariant metric of Hol($G$) conformal to the product invariant metrics of $G$ and Aut($G$). (motivated by $G=\mathbb{R}$, as an example of this situation)

2. For what type of Lie groups, $G$ is a totally geodesic submanifold of its holomorph?

And finally: can one write the lie algebra of Hol($G$) in term of Lie algebras of $G$ and $Aut(G)$ and also the natural action of $Aut(G)$ on the Lie algebra of $G$?