Character of induced representation: explicit formula I'm trying to work through Exercise 7.1 in Serre's "Linear Representations of Finite Groups" but I'm having trouble finishing my proof. The problem reduces to the following.
Let $N\subset H$ be a normal subgroup of a finite group $H$, and $V$ a $\mathbb{C}[H]$-module such that no nonzero vector is fixed by the whole of $N$. Let $\chi$ denote the corresponding character and $\pi: H\twoheadrightarrow H/N$ the projection. Prove that for any $t\in H/N$, 
$$ \sum_{\pi(s)=t} \chi(s) = 0. $$
When $t=1$ I can prove this, since we can decompose $V$ into irreducible $N$-representations and none of these can be the trivial one; thus we can use orthogonality of characters to finish, by inner-producting $\chi$ with the trivial character. However if $t\neq 1$ I can't see how to finish. I would try to inner-product $\chi$ with the function $f:H\to \mathbb{C}$ that sends $\{s \mid \pi(s)=t\}$ to 1 and the rest to 0, but this isn't (necessarily) a class function!
EDIT: I should clarify that the title refers to using $\pi: H\twoheadrightarrow (H/N)=: G$ to induce a $G$-representation $Ind_H^G(V)$. (We don't need to assume no nonzero vector of $V$ is fixed by the whole of $N$). Then Serre asks to prove its character shall be, for $t\in G$,
$$ t \mapsto \frac{1}{|N|}\sum_{\pi(s)=t} \chi(s). $$
 A: Let
$$\rho: H \rightarrow \mathrm{GL}(V)$$
denote the corresponding representation. Let
$$e_N:= \frac{1}{|N|} \sum_{n \in N} \rho(n) \in \mathrm{End}(V)$$
The endormorphism $e_N$ is just the projection of $V$ onto the $N$-invariant subspace, so, by assumption, $e_N = 0$ since $V^N = 0$.
We have $\chi(s) := \mathrm{Tr}(\rho(s))$. The elements of $H$ such that $\pi(s) = t$ are just the cosets (left or right since $N$ is normal) of some element in $H$, which by abuse of notation we can also call $t$. Thus
$$\begin{aligned} \sum_{\pi(s) = t} \chi(s) = & \
\sum_{n \in N} \chi(tn) \\
= & \ \sum_{n \in N} \mathrm{Tr}(\rho(t) \rho(n)) \\
= & \ \mathrm{Tr} \left( \rho(t) \sum_{n \in N} \rho(n) \right) \\
= & \ \mathrm{Tr} \left( \rho(t) |N| e_N \right)= \mathrm{Tr}(0) = 0.
\end{aligned}$$
ps. One usually reserves the notation $\mathrm{Ind}^{G}_{H}(V)$ to be the induction of a representation $V$ from a subgroup $H$ to $G$. Does Serre really use that notation? (Looks up bootleg copy of Serre -  he does not.) You should probably avoid it as it will cause confusion. The character turning up here is the character of $H/N$ acting on $V^N$. That's also easy enough to see from the same calculation above, since in general we get
$$\frac{1}{|N|} \sum_{\pi(s) = t} = \mathrm{Tr} \left( \rho(t) e_N \right)$$
which is just the trace of $t \in G/H$ acting on $V^N$.
