= Solution
If a <group representation> of $G$ has <character> $\chi$, its <restriction of a representation> to $H$ has character $\chi|_H$. For an $H$-representation $V$ with character $\psi$, the <induced representation> is $\mathbb C[G]\otimes_{\mathbb C[H]}V$, with $G$ acting on the first factor. Decomposing into coset copies of $V$ shows that its <character> is
$$
(\operatorname{Ind}_H^G\psi)(g)=\frac1{|H|}\sum_{\substack{x\in G\\x^{-1}gx\in H}}\psi(x^{-1}gx).
$$
Indeed only cosets fixed by $g$ contribute to the <trace>, and each coset has $|H|$ representatives.
The <Frobenius reciprocity> says $\langle\operatorname{Ind}_H^G\psi,\chi\rangle_G=\langle\psi,\operatorname{Res}_H^G\chi\rangle_H$. Directly, with the usual <character inner product>,
$$
\begin{aligned}
\langle\operatorname{Ind}\psi,\chi\rangle_G
&=\frac1{|G||H|}\sum_{x\in G}\sum_{h\in H}\psi(h)\overline{\chi(xhx^{-1})}\\
&=\frac1{|H|}\sum_{h\in H}\psi(h)\overline{\chi(h)}.
\end{aligned}
$$
The first equality substitutes $g=xhx^{-1}$ in the finite sum, and the second uses conjugacy invariance. This proves the theorem.
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