Induced representation from trivial representation on normal subgroup factors through regular representation of quotient group

Statement
Suppose $$H$$ is a normal subgroup of a group $$G$$. Denote by $$\psi$$ the trivial representation of $$H$$. Denote by $$\rho_{G/H}$$ the regular representation of $$G/H$$ and by $$q:G \to G/H$$ the quotient map. Then, the induced representation from $$H$$ to $$G$$ of $$\psi$$ factors via $$q$$ to $$\rho_{G/H}$$, i.e.,:

$$\operatorname{Ind}_H^G \psi = \rho_{G/H} \circ q$$

Related facts

 * Induced representation from regular representation of subgroup is regular representation of group
 * Induced representation from trivial representation of subgroup is permutation representation for action on coset space