 Index-two character restriction dichotomy

Let $H◃G$ have index two, and let $\lambda$ be the nontrivial linear character of $G/H$ inflated to $G$. For an irreducible character $\chi$ of $G$, either ${\mathrm{Res}}_H^G\chi$ is irreducible and $\chi \ne \chi \lambda$, or the restriction is a sum of two distinct irreducible characters and $\chi =\chi \lambda$.