  The fusion rules in $\mathrm{Rep}_f D(G)$ for a finite group $G$ can be
computed in terms of character inner products. Using an explicit formula for
these fusion rules, we show that $\mathrm{Rep}_f D(G)$ is multiplicity free for
two infinite families of finite groups: the Dihedral groups and the Dicyclic
groups. In fact, we will compute all fusion rules in these categories.
Multiplicity freeness is a desired property for modular tensor categories,
since it greatly simplifies the computation of $F$-matrices. Furthermore, we
observe that the fusion rules for Dihedral groups $D_{2n}$ with $n$ odd are
extremely similar to the fusion rules of Type $B$ level $2$ fusion algebras of
Wess-Zumino-Witten conformal field theories. Moreover, we give a proof of the
fusion rule formula by using Mackey theory.
