Induced class function from normal subgroup is zero outside the subgroup

Statement

Suppose $G$ is a finite group and $H$ is a Normal subgroup (?) of $G$. Suppose $\theta$ is a Class function (?) on $H$ with values in a field $k$. Then, the Induced class function (?) $\operatorname{Ind}_H^G \theta$ takes the value zero at all elements not in $H$.

Related facts

• Induced class function from conjugacy-closed normal subgroup is index of subgroup times class function inside the subgroup and zero outside the subgroup: A conjugacy-closed normal subgroup is a normal subgroup where two elements of the subgroup conjugate in the whole group are also conjugate in the subgroup. In particular, any direct factor and any central factor is conjugacy-closed.