A character of an induced representation I want a help to solve the following exercise from the book, Representation Theory, by Fulton and Harris.
Exercise 3.19 (p.34)
Let $H$ be a subgroup of a finite group $G$. Let $W$ be a representation of $H$.
If $C$ is a conjugacy class of $G$, and $C \cap H$ decomposes into conjugacy classes $D_1 , \dots, D_r$, the value of the character of $\mathrm{Ind} W$ on $C$ is
$$\chi_{\mathrm{Ind}W}(C)=\frac{|G|}{|H|} \sum_{i=1}^r \frac{|D_i|}{|C|} \chi_W(D_i).$$
I am supposed to rewrite the equation
$\chi_{\mathrm{Ind}W}(g) = \sum_{g \sigma=\sigma} \chi_W(s^{-1} gs)$, where $s \in \sigma$ arbitrary to get the above equality but I don't know where to start.
I appreciate any help.
 A: I am unable to think of a good way to reduce my ideas to mere hints. So here's a full solution.
Let $H\le G$ be a subgroup, $W$ an $H$-rep, $C$ a conjugacy class of $G$, and $C\cap H=\bigsqcup_{i=1}^r D_i$ its conjugacy class partition in $H$. The Frobenius formula states (for a transversal $~T$ of $G/H$)
$$\chi_{{\rm Ind}W}(x)=\sum_{t\in T}\chi_W(t^{-1}xt),$$
where $\chi_W|_{G-H}=0$ by fiat. This formula can be derived via ${\rm Ind}(W)\cong\bigoplus_{t\in T}tW$ and using the fact that ${\rm tr}(A)=\sum_i\langle Ae_i,e_i\rangle$ given an inner product and orthonormal basis $\{e_i\}$ wrt it. Further we know that $G$ can be partitioned into $|H|$ disjoint transversals as $G=\bigsqcup_{\ell=1}^{|H|}T_\ell$; identify the cosets of $H$ with rows in a table, and list out the elts of each coset in the specified row arbitrarily, then the columns will each be a transversal for $G/H$. Now we have the following computation:
$$\begin{array}{ll} \chi_{{\rm Ind}W}(C) & =\frac{1}{|C|}\sum_{c\in C}\chi_{{\rm Ind}W}(c) \\
& = \frac{1}{|C|}\sum_{c\in C}\sum_{t\in T}\chi_W(t^{-1}ct) \\ & =\frac{1}{|C|}\sum_{c\in C}\frac{1}{|H|}\sum_{\ell=1}^{|H|}\sum_{t\in T_{\Large\ell}}\chi_W(t^{-1}ct) \\ & = \frac{1}{|C||H|}\sum_{c\in C}\sum_{g\in G}\chi_W(g^{-1}cg) \\ & =\frac{1}{|H||C|}\sum_{i=1}^r\chi_W(D_i)\cdot\#\{(c,g,x)\in C\times G\times D_i:g^{-1}cg=x\} \\ & =\frac{1}{|H||C|}\sum_{i=1}^r\chi_W(D_i)\sum_{\substack{g\in G \\ x\in D_{\large i}}}\underbrace{\#\{c\in C:c=gxg^{-1}\}}_{=1} \\ & =\frac{|G|}{|H|}\sum_{i=1}^r\frac{|D_i|}{|C|}\chi_W(D_i).\end{array}$$
