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.