Suppose $H < K < G$ are finite groups and $G$ acts primitively by (right) multiplication on the set $\Gamma = G/K$ of (right) cosets of $K$ in $G$, and $K$ acts primitively on the set $\Delta = K/H$ of cosets of $H$ in $K$. Let's assume that $H$ is core-free in $K$ and $K$ is core-free in $G$, so these are, indeed, permutation groups.
I suspect there is a natural way to write the action of $G$ on the set $\Omega = G/H$ in terms of (a composition of) the actions $G$ on $\Gamma$ and $K$ on $\Delta$. In other words, I think the action of $G$ on $\Omega$ could be viewed as (1) permuting the cosets of $H$ which lie within a single $K$-coset, followed by (2) permuting the cosets of $K$. So it seems there is a wreath product underlying this, but I'm having trouble writing it down.
Here's what I have so far. Perhaps someone who knows more group theory can tell me what's right or wrong with it:
For each $g\in G$, we have the action $g : Hy \mapsto Hyg$. Suppose $g = kx$, for some $k \in K$ and some $x \in G$, where $x$ is a $K$-coset representative. Then the action of $g$ "factors through" the action of $k$ as follows:
$g: Hy \mapsto Hyk \mapsto Hykx$
Now, since we assumed these are permutation groups, we have $K\hookrightarrow Sym(\Delta)$ and $G\hookrightarrow Sym(\Gamma)$. Let $\mathcal K$ and $\mathcal G$ be the images of $K$ and $G$ under these embeddings.
Is the action of $G$ on the set $\Omega=G/H$ somehow related to the wreath product
$\mathcal K^{\Gamma} \rtimes \mathcal G \; ?$
(Incidentally, I'm fairly certain I don't need the primitivity assumptions, but in my application I happen to know that $H$ is maximal in $K$ and $K$ is maximal in $G$.)
Update: Professor Holt's answer below is perfectly clear, but I recently came across this nice article by Cheryl Praeger describing in detail exactly what I had in mind.