  In this paper, we give the decomposition into irreducible characters of the
restriction to the wreath product $\mathbb{Z}_{p-1} \wr \mathfrak{S}_w$ of any
irreducible character of $(\mathbb{Z}_p \rtimes \mathbb{Z}_{p-1}) \wr
\mathfrak{S}_w$, where $p$ is any odd prime, $w \geq 0$ is an integer, and
$\mathbb{Z}_p$ and $\mathbb{Z}_{p-1}$ denote the cyclic groups of order $p$ and
$p-1$ respectively. This answers the question of how to decompose the
restrictions to $p$-regular elements of irreducible characters of the symmetric
group $\mathfrak{S}_n$ in the $\mathbb{Z}$-basis corresponding to the $p$-basic
set of $\mathfrak{S}_n$ described by Brunat and Gramain in [1]. The result is
given in terms of the Littlewood-Richardson coefficients for the symmetric
group.
