Character of induced representation is induced class function of character

Statement

Suppose $H$ is a subgroup of finite index in a group $G$. Suppose $\varphi$ is a finite-dimensional linear representation of $H$. Let $\chi$ be the character of $\varphi$. Then, we have:

character of the induced representation $\operatorname {Ind} _{H}^{G}(\varphi )$ = induced class function $\operatorname {Ind} _{H}^{G}\chi$