representation of a group and its center

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Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \otimes V^*$ for each (representative of isomorphism class of) simple object of $C$.

I want to show that $D=\mathrm{Rep}(G/Z(G))$.

First I thought that for an irreducible representation $V$, we have $V \otimes V^* \cong \mathrm{Hom}(V, V)\cong k$, where $k$ is a base field. But it seems that this argument is wrong by considering dimensions.

I appreciate any help.