Conjugacy-class formula for an induced character (${\mathrm{Ind}}_H^G\chi (g)=\frac{|C_G(g)|}{|H|}{\sum }_{h\in g^G\cap H}\chi (h)$)

For a finite group $G$, a subgroup $H$, and a character $\chi$ of $H$, each element $h$ of $g^G\cap H$ has exactly $|C_G(g)|$ preimages under $x\mapsto x^{-1}gx$. This follows because one such preimage $x_0$ has full fiber $C_G(g)x_0$. Grouping the character formula for an induced representation by these fibers proves the formula. Grouping again by conjugacy classes of $H$ replaces each class by its size times its character value.