Let $G,H$ be groups with an action of $H$ on $G$, meaning a group homomorphism $H\to Aut(G)$, and let $S\le H$ be a subgroup, not necessarily normal.
Consider the semidirect products $G \rtimes H$ and $G \rtimes S$, formed respectively with the action above, and its restriction on $S$. Now the latter is a subgroup of the former. What can we say in general about the relationship between the two products, are there known results?
For example, consider the cosets of the inclusion $S\to H$ and the cosets of the inclusion $G\rtimes S\to G\rtimes H$. Are they isomorphic?