Let $(M, g)$ be a Riemannian manifold. I say $\lambda\in\mathbb R^\times$ is a scale factor of $M$ if $(M, g)$ is isometric to $(M, \lambda g)$. Composing isometries multiplies the scale factor, so there's subgroup $G=\{\lambda:\lambda\text{ is a scale factor of } M\}\leq\mathbb R^\times$ associated to this manifold which I call the group of scale factors of $M$. I want to ask: what subgroups of $\mathbb R^\times$ are achievable as groups of scale factors of some manifold?
If $M$ is compact, scaling by a factor other than $\pm 1$ changes the volume, so the group of scale factors is either trivial or $\{\pm 1\}$. Using de Rham cohomology we can prove that $\mathbb{CP}^{2n}$ has no orientation-reversing homeomorphisms into itself, and in particular no isometries thus its group is trivial. The sphere $\mathbb S^n$ has scale $-1$ automorphisms e.g. $(x_1, \dots, x_{n+1})\mapsto (-x_1, x_2, \dots, x_{n+1})$ thus its automorphism group is $\{\pm 1\}$.
This question asks about manifolds whose scale factors are the whole $\mathbb R^\times$ (such as $\mathbb R^n$) and one answer finds several examples. I think there should be an easy example of a manifold with group $\mathbb R_{>0}^\times$, but the question for any group other than these four that have been mentioned is way harder.