A corollary of the Mostow-Palais theorem is that every homogeneous space for a compact group is a linear group orbit. In other words, if $ H $ is a closed subgroup of a compact group $ K $ then there exists some representation $ \pi: K \to GL(V) $ and $ v \in V $ such that the orbit $$ \mathcal{O}_v=\{ \pi(k)v: k \in K\} $$ is diffeomorphic to $ K/H $ (the stabilizer of $ v $ is exactly $ H $).

Is the converse true; does every compact linear group orbit admit a transitive action by a compact group? In other words, if $ G $ is a group, $ \pi: G \to GL(V) $ a representation, $ v \in V $, and the stabilizer $$ \mathcal{O}_v=\{ \pi(g)v: g \in G \} $$ is compact, then must there exist some compact group $ K $ acting transitively on the manifold $ \mathcal{O}_v $?