Inner tensor and restriction

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 $G$?

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What do you mean by $\chi\otimes\pi$? One tensors representations with representations and multiplies characters with characters, but one does not mix the two. –  Mariano Suárez-Alvarez Jan 27 '12 at 9:05
What do you mean? $\mathbb{C}$ acts on every complex vector space. –  plusepsilon.de Jan 27 '12 at 9:44
@late_learner: you should just verify the left hand side and the right hand side are equal at every element of N. However, you should be careful that π is actually a linear character of G, not just the character of a representation, otherwise your χ⊗π will not be a homomorphism. –  Jack Schmidt Jan 27 '12 at 14:32