Action of wreath product on Cartesian product

Definition
Suppose both $$G$$ and $$H$$ are being viewed along with specific group actions of each, i.e., they are both being viewed as groups of permutations, with $$G$$ acting on a set $$A$$ and $$H$$ acting on a set $$S$$. Then, $$G \wr H$$ comes equipped with a natural action on the Cartesian product $$A \times S$$ as follows.

First, recall that $$G \wr S$$ is the external semidirect product $$G^S \rtimes H$$. To specify an action of this, we will specify how $$G^S$$ and $$H$$ act; then, via the equivalence of internal and external semidirect product, we would have defined the action of a generating set and hence of all of $$G \wr H$$. Here is the action:


 * $$G^S$$ acts on $$A \times S$$ as follows: An element of $$G^S$$ is a function $$f:S \to G$$. For an element $$(a,s)$$, the action of $$f$$ on $$(a,s)$$ is obtained as $$(f(s) \cdot a, s)$$ where $$f(s) \cdot a$$ is the action of the element $$f(s) \in G$$ on $$A$$. In other words, it permutes each of the fibers according to the $$f$$-value on that fiber.
 * $$H$$ acts on $$A \times S$$ as follows: An element $$h \in H$$ sends $$(a,s) \in A \times S$$ to $$(a,h \cdot s)$$ where $$h \cdot s$$ denotes the image of $$s$$ under the action of $$h$$.

Note that we need to verify a compatibility condition to show that this is well defined.