The characters of groups of type $X\wr\mathbb Z_p$

The irreducible complex characters of the groups $G\wr\mathbb Z_p$ are calculated, where $p$ is a prime and $G$ is a finite group with known characters table. As a consequence, we get a simple inductive method to find the characters tables of the Sylow $p$-subgroups of the symmetric groups. In particular, it is proved that the values of irreducible complex characters of the Sylow $2$-subgroups in such groups are rational which solves the problem 15.25 from “Kourovka Notebook”.