$\DeclareMathOperator{\Ind}{Ind}$Let $H$ be a subgroup of finite index of a group $G$. Let $\chi: G \rightarrow \mathbb C^{\ast}$ be a character of $G$, and $(\pi,V)$ a representation of $H$. Is it true that
$$\Ind(\pi \otimes \chi|_H) \cong \Ind(\pi) \otimes \chi$$
as representations of $G$?
The space of the left hand side consists of all functions $f: G \rightarrow V$ such that $f(hg) = \pi(h) \chi(h)f(g)$, on which $G$ acts by right translation. The space of the right hand side consists of all functions $f: G \rightarrow V$ such that $f(hg) = \pi(h)f(g)$, on which $G$ acts by $g \cdot f(x) = \chi(g)f(xg)$.
I have seen notes which identify these spaces, but are they really the same?