Suppose $\pi:P\rightarrow M$ is a principal $G$ bundle.
Let $H$ be a Lie group acting freely and properly on $P$ and on $M$ so that $P/H$ and $M/H$ are manifolds. Further assume this action is such that it defines a map $P/H\rightarrow M/H$.
Is it always true that the induced map $P/H\rightarrow M/H$ is also a principal $G$ bundle?
I think it is true, may be with some extra conditions.
Can some one please say what extra conditions (if at all) I need to confirm $P/H\rightarrow M/H$ is a principal $G$ bundle?