= Irreducible characters across an index-two subgroup
{title2=$\operatorname{Res}_H^G\operatorname{Ind}_H^G\psi=\psi+\psi^g$}
An index-two subgroup is normal. Conjugation by an outside element gives an involution on its irreducible characters. If $\psi^g\ne\psi$, the squared norm of induction is one, so induction is irreducible of twice the degree and restriction is $\psi+\psi^g$. If $\psi^g=\psi$, the squared norm is two, so induction has exactly two distinct constituents, each once. Each restricts to a representation containing $\psi$, and their total degree is twice that of $\psi$, forcing both degrees to equal that of $\psi$. This proves the fixed-orbit versus two-element-orbit classification without assuming extension of $\psi$ in advance.
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