Let $X$, $Y$ be sets and let G, H be groups which act on $X$, $Y$ (respectively). Denote the set of functions from $X$ to $Y$ by $X^Y$. I will use $f$ for functions in $X^Y$ and $g, h$ for elements of $G, H$, respectively.


As I understand it, the Polya Enumeration Theorem allows us to count the number of orbits of functions in $X^Y$ up to equivalence via the action of $G$ by:


$ g \cdot f(x) = f(x ^ g)$ for all $x \in X$.


There is also a generalization of Polya Enumeration Theorem (Frank Harray and Ed Palmer call it the Power Group Enumeration Theorem) which allows us to count functions in $X^Y$ up to equivalence via the following action of the direct product $G \times H$:


$(g, h) \cdot f(x) = h \cdot f(x ^ g)$ for all $x \in X$.


I want to ask whether the orbits of the wreath product $G \wr H = G \ltimes H^d$ (where $d = |X|$ is the degree of $G$) can be counted with respect to the following action


$(g, (h\_1, \dots, h\_d)) \cdot (y\_1, \dots, y\_d) = (h\_1 \cdot y\_{1^{g^{-1}}}, \dots, h\_d \cdot y\_{d^{g^{-1}}})$.



Is there a generalisation of the Polya Enumeration Theorem to actions of wreath products?
A good starting point with some useful references is 


[Enumeration under two representations of the wreath product](https://link.springer.com/content/pdf/10.1007/BF02392038.pdf) by Palmer and Robinson.


