Induced class function from normal subgroup is zero outside the subgroup

Statement
Suppose $$G$$ is a finite group and $$H$$ is a fact about::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 $$\operatorname{Ind}_H^G \theta$$ 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.