If $G$ is a finite group and its conjugacy classes are known, can the conjugacy classes of the wreath product $G \wr S\_n \cong G^n \rtimes S\_n$ be determined?



Determining the conjugacy classes of a wreath product $G \wr S_n$
This is indeed handled in Section 4.2 of James and Kerber's book on Representations of the Symmetric group.


We also considered this problem in Section 2 our paper describing a computer algorithm for computing conjugacy class representatives in permutation groups.


J. Cannon and D. Holt, Computing conjugacy class representatives in permutation groups,
J. Algebra 300 (2006), 213--222.


I can cut and paste the main bit of theory you need. The problem easily reduces to finding those classes that map onto a specific element of $S\_n$.


Let $A$ be any group, let $P$ be a permutation group acting
on the set $\{ 1\ldots d\}$, and let $W := A \wr P$.
Then $W$ is a semidirect product of $A^d$ by $P$, and elements
of $W$ have the form $(g,x)$ with $g \in P$, $x \in A^d$, and
$x = (x\_1,\ldots,x\_d)$ with $x\_i \in A$. In general, for $x \in A^d$,
we denote the $i$-th component of $x$ by $x\_i$.
The action of $P$ on $A^d$ in $W$ is given by:
$$(g,1)^{-1}(1,(x\_1,\ldots,x\_d))(g,1) = (1,(x\_{1^{g^{-1}}},\ldots,
x\_{d^{g^{-1}}})).$$


Hence, for $g \in P$, $x,z\in A^d$, we have
\begin{equation}\label{conjeqn}
(1,z)^{-1}(g,x)(1,z) = (g,y), \ \mathrm{where}\ 
y\_i = z^{-1}\_{i^{g^{-1}}}x\_iz\_i \ \mathrm{for}\ 1 \le i \le d.
\end{equation}


$\textbf{Theorem}$ With the above notation,
let $x^{(1)},\ldots,x^{(k)}$ be representatives of the conjugacy classes of $A$.
Fix an element $g \in P$, and let $r\_1,r\_2,\ldots,r\_s$ be representatives of
the cycles of $g$ in its action on $\{ 1\ldots d\}$. Then
$$
R := \{\,(g,x) \mid x\_i = 1 \ \mathrm{for}\ i \not\in \{r\_1,\ldots,r\_s\},\,
 x\_i \in \{x^{(1)},\ldots,x^{(k)}\}\ \mathrm{for}\
i \in \{r\_1,\ldots,r\_s\}\,\}
$$
is a set of representatives of the $A^d$-classes of elements of $W$
of the form $(g,x)$ for $x \in A^d$.


The size of the $A^d$-class $R\_{gx}$ containing $(g,x)$ is the product of the sizes of the classes of the $x\_i$ in $A$ with $i \in \{r\_1,\ldots,r\_d\}$ with the lengths of the cycles of $g$ that contain $r\_1,\ldots,r\_d$.


The above describes the $A^d$-classes of $W$; i.e. the orbits under conjugation by the subgroup $A^d$ of $W$. To get the classes of $W$ itself, first of all we only consider those $R\_{gx}$ as $g$ ranges over a set of class representatives of $P$. Then, for each such $g \in W$, the $A^d$-class representatives $R\_{gx}$ are fused under the action of $C\_P(g)$. This action is the same as the action of $C\_P(g)$ on the cycles of $g$, so is easy to calculate.


Then the size of a $W$-class $R\_{gx}$ is the product of the size of the class of $g$ in $P$ with the sum of the the sizes of the $A^d$-classes $R\_{gx'}$ that lie in the orbit under the action of $C\_P(g)$. 


