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In this paper, we give the decomposition into irreducible characters of therestriction to the wreath product $\mathbb{Z}_{p-1} \wr \mathfrak{S}_w$ of anyirreducible 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 therestrictions to $p$-regular elements of irreducible characters of the symmetricgroup $\mathfrak{S}_n$ in the $\mathbb{Z}$-basis corresponding to the $p$-basicset of $\mathfrak{S}_n$ described by Brunat and Gramain in [1]. The result isgiven in terms of the Littlewood-Richardson coefficients for the symmetricgroup.

Restriction of characters to subgroups of wreath products and basic sets for the symmetric group