Let $G$ be a finite group, $N$ its normal subgroup, and $\pi:N\to{\mathcal B}(X)$ a unitary representation of $N$ in a Hilbert space $X$. Consider the induced representation $\pi':G\to{\mathcal B}(L\_2(F,X))$, where $F=G/N$. 


Let us extend the representations $\pi$ and $\pi'$ to the group algebras: 
$$
\pi:{\mathbb C}[N]\to{\mathcal B}(X),\qquad \pi':{\mathbb C}[G]\to{\mathcal B}(L\_2(F,X)).
$$


Question:



> 
> is it true that for all elements $a\in {\mathbb C}[N]$ the norms of the operators $\pi(a)\in {\mathcal B}(X)$ and $\pi'(a)\in {\mathcal B}(L\_2(F,X))$ coincide
>  $$
> ||\pi(a)||=||\pi'(a)||\quad ?
> $$
> 
> 
> 


I was sure, that this is true for all $G$, $N$ and $\pi$ (and not only for finite groups), but recently I found unexpectedly, that I can prove this only for the case of $G=N\times F$. Is it possible that there exists a counterexample for $G\ne N\times F$?



Does the induced representation preserve norm?
No. Let $G=S\_3$ generated by 3-cycle $x$ and involution $y$. Let $N=A\_3$. Let $ x$ act on $X=\mathbb C$ by multiplication by the third root of unity $\omega$. Let $a=1+ix$. Then $\pi(a)$ has norm $|1+i\omega|$ but $\pi'(a)y=(1+i\omega^2)y$ showing $\pi'(a)$ has larger norm.


