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

Statement

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 $G$.

In particular, any Direct factor (?) or any Central factor (?) is a conjugacy-closed normal subgroup, so the above applies.

Related facts

• Induced class function from normal subgroup is zero outside the subgroup