Let $G \le S_n$ and $H \le G$ be subgroup. Let $G$ acts on cosets of $H$ in $G$ and it forms a block system ($\forall \sigma \in G, S^{\sigma} = S $), where $S = \{Hg_1, ...,Hg_k\}$.
Question : What is the condition on $H$ such that action $\phi: G \times S \mapsto S$ is primitive (no nontrivial blocks)?