**Edit:** According to the comment of Ycor we remove the phrase "Naturally arises from a left invariant metric'

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. We fix an orientation for $G$. We denote by $H$ the group of all orientation preserving diffeomorphisms $f$ of $G$ such that for every $X\in \mathfrak{g}$ we have $f^* X$ belong to $\mathfrak{g}$.

Is there a natural Lie group structure for $H$ which contains $G$ as a closed normal subgroup? How can the constant curvature of $H$ and also its Lie algebra be related to those objects for $G$?

As a simplest example we get the group of affine linear maps $x\mapsto ax+b,\quad a>0$ of the real line when $G=\mathbb{R}$, a model of Poincare upper plane.