= Solution
The induced representation is
$$
\operatorname{Ind}_H^G(V)=\mathbb C[G]\otimes_{\mathbb C[H]}V,
$$
with $G$ acting by left multiplication on the first tensor factor. If $R$ is a set of representatives for the left cosets $G/H$, then as a vector space it is $\bigoplus_{r\in R}r\otimes V$.
The operator $g$ permutes these summands. A summand indexed by $xH$ contributes to the trace exactly when $gxH=xH$, equivalently when $x^{-1}gx\in H$, and its contribution is $\chi_V(x^{-1}gx)$. Accounting for the $|H|$ representatives of each coset gives
$$
\boxed{
\chi_{\operatorname{Ind}_H^G V}(g)
=\frac1{|H|}
\sum_{\substack{x\in G\\x^{-1}gx\in H}}
\chi_V(x^{-1}gx).}
$$
For a $G$-representation $W$, restrict $W$ to $H$ in the left-hand induced representation. Define
$$
\Phi:\mathbb C[G]\otimes_{\mathbb C[H]}(W\otimes V)
\longrightarrow
W\otimes\bigl(\mathbb C[G]\otimes_{\mathbb C[H]}V\bigr)
$$
by
$$
\Phi\bigl(x\otimes(w\otimes v)\bigr)
=xw\otimes(x\otimes v).
$$
It is well-defined because both representatives of the balancing relation give
$$
xhw\otimes(x\otimes hv).
$$
It is $G$-equivariant, and its inverse is
$$
w\otimes(x\otimes v)\longmapsto
x\otimes(x^{-1}w\otimes v).
$$
Consequently
$$
\boxed{
\operatorname{Ind}_H^G(\operatorname{Res}_H^G W\otimes V)
\cong W\otimes\operatorname{Ind}_H^G(V).}
$$
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