In this paper we prove that each $g$-natural metric on a linear frame bundle $LM$ over a Riemannian manifold $(M,g)$ is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define $g$-natural metrics on the orthonormal frame bundle $OM$ and we prove the same invariance result as above for $OM$. Hence we see that, over a space $(M,g)$ of constant sectional curvature, the bundle $OM$ with an arbitrary $g$-natural metric $\tilde{G}$ is locally homogeneous.