Let $\pi$ resp. $\chi$ be a finite dimensional representation resp. 1-diml. rep. of a finite group.
We define $\chi \otimes \pi (g) = \chi(g) \pi(g)$ as a rep of $G$.
For $N$ subgroup, does hold $$Res_N \chi \otimes \pi = Res_N \chi \otimes Res_N \pi?$$
If not, what happens, if $N$ is normal inside $G$?