Let $G$ be a finite group and $K$ a field. Let $M$ be a $K[H]$-module and let
$$\operatorname{ind}_H^G M := \lbrace f : G \to M : f(gh) = h^{-1}f(g)\ \text{for all}\ g\in G, h\in H\rbrace \subseteq \operatorname{Hom}(G,M)$$
be the $K[G]$-module induced by $M$.
Define by $g_i\gamma_M(M) := \lbrace f \in \operatorname{ind}_H^G M : f(G\setminus g_i H) = 0\rbrace$. It can be shown that
$$\operatorname{ind}_H^G M = \bigoplus_{i=1}^r g_i \gamma_M(M)$$
with $(g_i)_i$ a system of coset representatives for $H$ in $G$ and $r := [G:H]$. I am trying to prove the following lemma:
If $\operatorname{char} K \nmid |H|$ then $$\chi_{\operatorname{ind}_H^G M}(g) = \frac{1}{|H|}\sum_{k \in G}\dot\chi_M(kgk^{-1})$$ where $\dot\chi_M(g) = \chi_M(g)$ for $g \in H$ and $\dot\chi_M(g) = 0$ for $g \in G\setminus H$.
Here are some thoughts:
Since $\operatorname{ind}_H^G M = \oplus_{i=1}^rg_i\gamma_M(M)$ we have
$$\chi_{\operatorname{ind}_H^G M}(g) = \sum_{i=1}^r \chi_{g_i\gamma_M(M)}(g).$$
If $g_{g_i\gamma_M(M)} : g_i\gamma_M(M) \to g_i\gamma_M(M)$ is the multiplication by $g$ map then
$$\chi_{\operatorname{ind}_H^G M}(g) = \sum_{i=1}^r Tr(g_{g_i\gamma_M(M)}) = Tr\left(\sum_{i=1}^r g_{g_i\gamma_M(M)} \right).$$
Since $G = \bigsqcup_{i=1}^r g_i H$, for each $g \in G$ there is a $\sigma_g \in S_r$ and $h_{g, i}\in H$ such that $gg_i = g_{\sigma_g(i)}h_{g, \sigma_g(i)}$. Given a $g_i \gamma_M(x_i) \in g_i\gamma_M(M)$ we have
$$gg_i\gamma_M(x_i) = g_{\sigma_g(i)}h_{g, \sigma_g(i)}\gamma_M(x_i) = g_{\sigma_g(i)}\gamma_{M}(h_{g, \sigma_g(i)}x_i) \in g_{\sigma_g(i)}\gamma_M(M),$$ (excuse the notation) so multiplication by $g$ seems to just permute the direct summands of $\operatorname{ind}_H^G M$.
Edit: by the direct sum decomposition of $\operatorname{ind}_H^G M$: given an $f \in \operatorname{ind}_H^G M$ we have
$$gf = \sum_{i=1}^r gg_i\gamma_M(x_i) = \sum_{i=1}^r g_{\sigma_g(i)}\gamma_M(h_{g, \sigma_g(i)}x_i)$$
Edit: Despite the reference given by BW. in the comments I'll continue here.
Since $M \cong g_i\gamma_M(M)$ as $K$-vector spaces ($\gamma_M$ is an injective linear map, hence preserves dimensions and $g_i \cdot -$ is an isomorphism) we have $\chi_{g_i\gamma_M(M)} = \chi_M$, so that
$$\chi_{\operatorname{ind}_H^G M}(g) = \sum_{i=1}^r \chi_M(g) = [G:H]\chi_M(g) = \frac{1}{|H|}\sum_{k \in G}\chi_M(kgk^{-1})$$
where the last step follows because $\chi_M(g) = \chi_M(kgk^{-1})$ for all $g,k \in G$.
I guess this last edit probably contains some severe logical errors.. but I can't see them!