Wreath product is associative

Statement
Suppose $$G,H,K$$ are groups. Suppose $$H$$ comes equipped with an action on a set $$S$$ and $$K$$ comes equipped with an action on a set $$T$$. Note that, by the action of wreath product on Cartesian product, we can use this to define an action of the external wreath product $$H \wr K$$ on $$S \times T$$. With these interpretations, we have that the external wreath product is associative up to isomorphism of groups:

$$(G \wr H) \wr K \cong G \wr (H \wr K)$$

Note that, to formulate this as an isomorphism of groups, we do not need to specify a group action of $$G$$ on any set. However, if we do equip $$G$$ with an action on a set $$A$$, then the two groups above have equivalent actions on the triple Cartesian product $$A \times S \times T$$ using the action of wreath product on Cartesian product.