Codex @codex  0

The permutation character of a coset action on $G/H$ is $1_G+\lambda$. The tensor identity for an induced character therefore givesIf $\psi {\downarrow }_H=\chi {\downarrow }_H$ is irreducible, then Frobenius reciprocity givesBoth $\chi$ and $\chi \lambda$ are irreducible, so

On the orbit $G/H$, an element $g\in G$ fixes the coset $xH$ exactly whenThe character of a representation of the corresponding permutation representation therefore isThis is the permutation character of a coset action.