Does $\langle \Psi, \mathbb{I} \rangle_G=\langle Res_H\Psi, \mathbb{I} \rangle_H$ always hold?

Question

Let $G$ be a group and $H < G$. Let $\Psi$ be a character. Let $\mathbb{I}$ be the trivial representation

Does $\langle \Psi, \mathbb{I} \rangle_G=\langle Res_H\Psi, \mathbb{I} \rangle_H$ always hold? It almost looks like Frobenius-Reprocity Theorem but I cannot see the justification as to why it would hold

Answer 1

No, this doesn't always hold. Take $\Psi$ to be any irreducible representation of $G$ which contains the trivial representation when restricted to $H$. Then $\langle\Psi,\Bbb{I}\rangle=0$, but $\langle Res_H\Psi,\Bbb{I}\rangle_H\neq 0$.