Equivalence of group representations if their restrictions are equivalent.

Question

Let $G=H\ltimes N$ be the semidirect product of a finite group $H$ and a finite abelian group $N$. Suppose that $\pi$ and $\eta$ are representations of $G$ such that $\pi|_{H\times\{e\}}\cong\eta|_{H\times\{e\}}$ and $\pi|_{\{e\}\times N}\cong\eta|_{\{e\}\times N}$. Can we conclude from this that $\pi\cong\eta$? It seems very logical at first glance but it goes wrong when I simply want to write it out.

Answer 1

There is a counterexample with $|H|=|N|=2$ (so $G = H \times N$). The character table of $G$ is

C1      1  1  1  1
C2      1 -1  1 -1
C3      1  1 -1 -1
C4      1 -1 -1  1

where $H$ is the union of the first two columns and $N$ is the union of the first and third columns.

Then the characters C1 + C4 and C2 + C3 are distinct but have the same restrictions to both $H$ and $N$.