= Solution
The <induced character> construction extends any <class function> $\psi$ on $H$ by
$$
\operatorname{Ind}_H^G\psi(g)=\frac1{|H|}\sum_{x\in G:\,x^{-1}gx\in H}\psi(x^{-1}gx).
$$
<Frobenius reciprocity> for class functions is $\langle\operatorname{Ind}_H^G\psi,\chi\rangle_G=\langle\psi,\operatorname{Res}_H^G\chi\rangle_H$. It follows directly by substituting this sum into the left <inner product>, writing $g=xhx^{-1}$, and using the conjugacy invariance of $\chi$. If $\psi$ is a <character>, its <inner product> with every restricted <irreducible character> is a nonnegative integer, a multiplicity in its complete reducible decomposition. Expanding in the <orthonormal> irreducible-character basis therefore makes $\operatorname{Ind}\psi$ a nonnegative integer sum of irreducible characters: \b[it is a <character> of $G$].
If $U$ affords $\psi$, a concrete <induced representation> is $\mathbb C[G]\otimes_{\mathbb C[H]}U$, with $G$ acting by left multiplication on the first factor. Choosing <coset> representatives decomposes it into $[G:H]$ copies of $U$. An element $g$ permutes these copies; only fixed <cosets> contribute <trace>, and the contribution from $xH$ is $\psi(x^{-1}gx)$. Summing fixed <cosets> gives exactly the displayed induced <class function>.
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