In Section 3.2 of Linear Representations of Finite Groups by Serre, the author gives a formula to compute the characters of product of finite groups (Abelian or non-Abelian).
First Serre defines the product of two groups.
Let $G_1$ and $G_2$ be two groups, and let $G_1 \times G_2$ be their product, that is, the set of pairs $\left(s_1, s_2\right)$, with $s_1 \in G_1$ and $s_2 \in G_2$. Then,
$$ \left(s_1, s_2\right) \cdot \left(t_1, t_2\right) = \left(s_1 t_1, s_2 t_2\right) $$
Then, Serre introduced the representation of the product as follows.
Now let $\rho^1 : G_1 \to GL\left(V_1\right)$ and $\rho^2 : G_2 \to GL\left(V_2\right)$ be linear representations of $G_1$ and $G_2$ respectively. We define a linear representation $\rho^1 \otimes rho^2$ of $G_1 \times G_2$ into $V_1 \otimes V_2$ by setting
$$\left(\rho^1 \otimes \rho^2\right)\left(s_1, s_2\right) = \rho^1 \left(s_1\right) \otimes \rho^2 \left(s_2\right)$$
Finally, comes the formula for computing the characters of the product.
$$\chi \left(s_1, s_2\right) = \chi_1 \left(s_1\right) \cdot \chi_2 \left(s_2\right)$$
In the same section, Serre introduces Theorem 10 which is as follows.
Theorem 10: (i) If $\rho^1$ and $\rho^2$ are irreducible, $\rho^1 \otimes \rho^2$ is an irreducible representation of $G_1 \times G_2$.
(ii) Each irreducible representation of $G_1 \times G_2$ is isomorphic to a representation $\rho^1 \otimes \rho^2$, where $rho^i$ is an irreducible representation of $G_i (i = 1, 2)$.
My question: Do we have similar results for semidirect and wreath products of non-Abelian finite groups?
My effort: I have found in Section 8.2 of Serre how to compute the characters when one of the group is Abelian. I have no clue how to extend the idea for non-Abelian group. In Representations of semidirect products, Akhil Mathew defines the irreps of semidirect products but it is not clear to me how to compute the character of the product.