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?
As written in the comments, what "cosets isomorphic" means is not clear. However, there is a natural map $f$ from $H/S$ to $G\rtimes H/G\rtimes S$ defined by sending $h\cdot S$ to $(1,h)\cdot G\rtimes S$.
What you can check is that $f$ is one-to-one and onto. Furthermore for any $h,h'$ in $H$ :
$$f(h\cdot(h'\cdot S))=h\cdot ( h'\cdot G\rtimes S)\text{.}$$
In other words $f$ is an isomorphism if $H/S$ and $G\rtimes H/G\rtimes S$ are given the structure of $H$-set (i.e. sets with an action of $H$) and if you don't assume $S$ normal in $H$ this is the only isomorphism you can have.