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 fact about::conjugacy-closed normal subgroup (i.e., $$H$$ is both a conjugacy-closed subgroup and normal subgroup) of $$G$$. Suppose $$\theta$$ is a fact about::class function on $$H$$ with values in a field $$k$$. Then, the fact about::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 fact about::direct factor or any fact about::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