$H$ and $K$ are groups, and $\Gamma$ is a set acted upon by H, while $\Delta$ is a set acted upon by $K$. Let $W := K \wr_\Gamma H$, the wreath product of $H$ and $K$. I have seen theorems stating that, ($H$ on $\Gamma$) and ($K$ on $\Delta$) are both transitive/faithful iff ($W$ on $\Delta \times \Gamma$) is a transitive/faithful action. There are also similar, though not identical results, for the action of $W$ on $\Delta^{\Gamma}$: the set of functions $f: \Gamma \rightarrow \Delta$, based upon the actions of the underlying groups.
Now, here is my question. Let $S$ := $H \ltimes_\phi K$, for some homomorphism $\phi: H \rightarrow \operatorname{Aut}(K)$. Are there similar such results relating the properties of S acting on some set built from $\Delta$ and $\Gamma$, such as their cartesian product, to the properties of ($H$ on $\Gamma$) and ($K$ on $\Delta$)?