Torsion-free connections on $G$-structures

We prove that for a group ${\mathrm{S}\mathrm{O}}_n(\mathrm{R})\subset G\subset {\mathrm{G}\mathrm{L}}_n(\mathrm{R})$, any $G$-structure on a smooth manifold can be endowed with a torsion free connection which is locally the Levi-Civita connection of a Riemannian metric in a given conformal class. In this process, we classify the admissible groups.