Systems of imprimitivity for wreath products

Let $G$ be an irreducible imprimitive subgroup of ${\mathrm{GL}}_n(\mathbb{F})$, where $\mathbb{F}$ is a field. Any system of imprimitivity for $G$ can be refined to a nonrefinable system of imprimitivity, and we consider the question of when such a refinement is unique. Examples show that $G$ can have many nonrefinable systems of imprimitivity, and even the number of components is not uniquely determined. We consider the case where $G$ is the wreath product of an irreducible primitive $H \leq \operatorname{GL}_d($\mathbb{F}$)$andtransitive$K \leq S_k$,where$n = dk$. We show that $G$ has a unique nonrefinable system of imprimitivity, except in the following special case: $d=1$, $n=k$ is even, $\mathrm{\mid }H\mathrm{\mid }=2$, and $K$ is a transitive subgroup of $C_2\wr S_{n\mathrm{/}2}$. As a simple application, we prove results about inclusions between wreath product subgroups.