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 $G\ne N\times F$?