If $Z$ is a subgroup of the center of G and $|G:Z|=m$, then $\chi^*(g)=m\chi(g)$ if $g\in Z$ where $\chi$ is a character of $Z$ and $\chi^*$ is the induced character of $G$.
Let $\phi$ be the representation of $Z$, and $\Phi$ be the induced representation. I am having a hard time of finding the relation between the trace of $\Phi$ and when it is restricted to $\phi$. What can we say about the traces? From the question it seems that the traces "differ" by $m$ and the idea of Schur's lemma will be helpful in this case.