Action of wreath product on function space

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 set $$A^S$$ of all functions from $$S$$ to $$A$$.

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 $$A^S$$ as follows: Given a function $$f: S \to G$$ and a function $$\theta:S \to A$$, define $$f \cdot \theta$$ as the function $$S \to A$$ given by $$s \mapsto f(s) \cdot \theta(s)$$, where we use the action of $$G$$ on $$A$$. One way of thinking of this is that the action is coordinate-wise or point-wise.
 * $$H$$ acts on $$A^S$$ as follows: Given an element $$h \in H$$ and a function $$\theta:S \to A$$, we define $$h \cdot \theta$$ as the map $$s \mapsto \theta(h^{-1} \cdot s)$$ where we use the action of $$H$$ on $$S$$ on the inside.