Let $\pi : M \to B$ be a smooth principal bundle with group $G$, where $M$ is an analytical (Fréchet, in my case) manifold, $B$ is a smooth (Fréchet) manifold and $G$ is a smooth (Fréchet) Lie group. Finally, let $S \subset M$ be an analytic submanifold of codimension one. What are "reasonable" hypotheses to ensure that $S$ descends to the quotient $B$ as an analytic submanifold?

Let $\pi : M \to B$ be a smooth principal bundle with group $G$, where $M$ is an analytic (Fréchet, in my case) manifold, $B$ is a smooth (Fréchet) manifold and $G$ is a smooth (Fréchet) Lie group. Finally, let $S \subset M$ be an analytic submanifold of codimension one. What are "reasonable" hypotheses to ensure that $S$ descends to the quotient $B$ as an analytic submanifold?