Let $G$ be a finite permutation group acting transitively and imprimitively on $\Omega$ with nontrivial block $\Delta \subset \Omega$. We can define a suitable action of the wreath product $G_{\Omega/\Delta} \wr G_\Delta = G_{\Omega/\Delta} \ltimes (G_\Delta)^{\Omega/\Delta}$ on $\Omega$, where $G_{\Omega/\Delta}$ and $G_\Delta$ are the permutation groups on $\Omega/\Delta$ and $\Delta$, respectively, which are induced by the action of $G$ on $\Omega/\Delta$, and by the action of the set stabilizer of $\Delta$ in $G$. We always have an embedding $G \hookrightarrow G_{\Omega/\Delta} \wr G_\Delta$ of permutation groups. I am looking for sufficient criteria under which this embedding is an isomorphism.
Of course a trivial criterion is the order of $G$, namely: $G \hookrightarrow G_{\Omega/\Delta} \wr G_\Delta$ is bijective if (and only if) $|G| = |G_{\Omega/\Delta}| \cdot |G_\Delta|^{|\Omega/\Delta|}$. I am looking for other criteria, which can be applied if I am not able to compute the order of $G$.