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 $G$?