Let $G$ be a compact Lie group acting on a compact manifold $M$. Let $S \subset M$ be a compact embedded submanifold. Then is $$G \cdot S = \{ g.s \in M : g\in G, s \in S \} $$ an embedded submanifold of $M$?
I know that this is true if $S$ is equal to a single point, however I could not find a general statement for when $S$ is a submanifold. If this is true, could you please provide a reference?