Irreducible characters across an index-two subgroup (${\mathrm{Res}}_H^G{\mathrm{Ind}}_H^G\psi =\psi +{\psi }^g$)

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.