Suppose $G$ is a finite group, and $H$ a subgroup. For an irreducible character $\chi$ of $G$, there is a central idempotent in the group algebra $\mathbb{C}[G]$: $$ e_\chi=\frac{\chi(1)}{|G|}\sum_{g\in G}\chi(g^{-1})g. $$
I write $e_H:=e_{1_H}=\frac{1}{|H|}\sum_{h\in H}h$ for the idempotent in $\mathbb{C}[H]$ corresponding to the trivial character on $H$. If $[G/H]$ denotes a complete set of left coset representatives of $H$ in $G$, then by construction the element $$ \sum_{g\in [G/H]}ge_Hg^{-1} $$ is central in $\mathbb{C}[G]$. Is there any way to somewhat explicitly extract which characters have their corresponding central idempotents in this linear combination, and/or their multiplicity? Just looking at the Mackey formula and since $ge_Hg^{-1}=e_{gHg^{-1}}$, my guess is it may be characters that are constituents of $\operatorname{Ind}^G_H(1_H)$, or something similar, but I am unsure.
