Suppose $H$ is a Hopf algebra over a field $K$ and $S$ is a ring which has a right $H$-action.
If ${}_HA$ is a pure left $H$-submodule, will $S\otimes_H A$ be a pure $S$-submodule ? If not, then what are the minimum conditions required (on $H$ or $S$) to make it so?