A relationship between character values of wreath products and the symmetric group

A relation between certain irreducible character values of the hyperoctahedral group [Formula: see text] ([Formula: see text]) and the symmetric group [Formula: see text] was proved by Lübeck and Prasad in 2021. Their proof is algebraic in nature and relies on methods from Lie theory. Using combinatorial methods, Adin and Roichman established a similar relation between certain character values of [Formula: see text] and [Formula: see text], where [Formula: see text] is an abelian group of order [Formula: see text], thereby generalizing the result of Lübeck-Prasad. Building on their work, we prove a further relation between certain irreducible character values of [Formula: see text] and [Formula: see text], again with [Formula: see text] an abelian group of order [Formula: see text].