= Solution
By the <Maschke theorem>, the restricted <representation> is a direct sum of irreducible $H$-<modules>. The <row orthogonality relations for a character table> identify the multiplicity of $\psi_i$ as $d_i=\langle\chi\mathord\downarrow_H,\psi_i\rangle_H$. Therefore
$$
\chi\mathord\downarrow_H=\sum_i d_i\psi_i,\qquad d_i\in\mathbb Z_{\geq0}.
$$
Using the preceding <character inner product> identity and the irreducibility of $\chi$,
$$
\sum_i d_i^2=\frac1{|H|}\sum_{h\in H}|\chi(h)|^2\leq\frac1{|H|}\sum_{g\in G}|\chi(g)|^2=\frac{|G|}{|H|}.
$$
This is the <character restriction norm bound>. When $[G:H]=2$, the nonzero sum of squares of integer multiplicities is either one or two. Hence \b[the restriction is irreducible, or is the sum of two distinct <irreducible characters>], each with multiplicity one. This proves the requested <index-two character restriction dichotomy> directly; normality of an index-two subgroup is a general <group theory> fact.
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