A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$.
Proposition: A finite group $G$ is perfect iff every $1$-dimensional complex representation of $G$ is trivial.
proof: First if $G$ is perfect, then every element $g \in G$ is of the form $[a_1,b_1] \dots [a_n, b_n]$, then for every $1$-dimensional complex representation $\pi$ of $G$, $\pi(g) = 1$, so $\pi$ is trivial.
Next we assume that every $1$-dimensional complex representation of $G$ is trivial; let $A:=G/G^{(1)}$, it is an abelian group so the irreducible complex representations of $A$ are $1$-dimensional, let $(\pi,V)$ one of them, and let $(\tilde{\pi},V)$ be the $1$-dimensional complex representation of $G$ defined by $\tilde{\pi}(g) \cdot v = \pi(gG^{(1)}) \cdot v$ (with $v \in V$ and $gG^{(1)} \in A$). Then by assumption $\tilde{\pi}$ is trivial, so a fortiori $\pi$ is trivial, and so $A$ is the trivial group and $G$ is perfect. $\square$
Definition: A fusion category (resp. ring) is perfect if every simple object of PFdim. $1$ is the trivial object.
By the previous proposition, the fusion category $Rep(G)$ is perfect iff $G$ is perfect.
Question: Is there a perfect integral fusion category $\mathcal{C}$ of PFdim $\equiv 2 [4]$?
Proposition: If $\mathcal{C} = Rep(G)$ is perfect then PFdim $\not \equiv 2 [4]$.
proof: We have to show that if $G$ is perfect then $\vert G \vert \not \equiv 2 [4]$. The following proof by contradiction is the development of this comment of Yves de Cornulier. If $\vert G \vert \equiv 2 [4]$ then let $τ$ be an element of order $2$ (it exists by Cauchy's theorem), and let $\rho$ be the permutation representation of the left action of $G$ on itself, then $\rho(τ)$ is a permutation of the form $(g_1,τg_1) \cdots (g_r,τg_r)$ with $r=|G|/2$; it follows that $sign(\rho(τ))=(−1)^{|G|/2}=−1$ by assumption. Conclusion $sign(\rho(G))=\{−1,1\}$, but this is impossible for a perfect group because $sign([a,b])=sign(a)^2sign(b)^2=1$. $\square$
Application: If the answer is no, then we get a necessary condition for some fusion rings to be categorifiable; more precisely, if a perfect integral fusion ring admits a PFdim $\equiv 2 [4]$ then it can't be categorifiable. Then the perfect integral fusion ring of PFdim. $210$ in this post would be non-categorifiable and so not the Grothendieck ring of a non-trivial maximal finite dimensional Kac algebra (i.e. Hopf ${\rm C}^*$-algebra), and so not the even part of a non-group maximal finite index depth $2$ irreducible subfactor.
