# Induced class function from conjugacy-closed normal subgroup is index of subgroup times class function inside the subgroup and zero outside the subgroup

Suppose $G$ is a finite group and $H$ is a Conjugacy-closed normal subgroup (?) (i.e., $H$ is both a conjugacy-closed subgroup and normal subgroup) of $G$. Suppose $\theta$ is a Class function (?) on $H$ with values in a field $k$. Then, the Induced class function (?) $\operatorname{Ind}_H^G \theta$ is as follows:
$\operatorname{Ind}_H^G \theta(x) = \lbrace\begin{array}{rl}[G:H]x, & x \in H \\ 0, & x \notin H \\\end{array}$
Here, $[G:H]$ is the index of $H$ in $G$.