 Solution

Because $[G:H]=2$, part (a) says that the squared norm of $\chi {\downarrow }_H$ is either one or two. It is one exactly when the restriction is irreducible, whereas it is two exactly when $\chi$ vanishes throughout $G\setminus H$. Thus condition (1) is equivalent to the existence in condition (2).

On $H$ we always have $\chi \lambda =\chi$, while outside $H$ we have $\chi \lambda =-\chi$. Therefore $\chi \lambda =\chi$ exactly when $\chi$ vanishes outside $H$. This proves all three equivalences in the index-two character restriction dichotomy: