In Chapter 15 Section 15.2.1 of Quantum Groups and Noncommutative Geometry, 2nd edition, the authors raised a question: can we reconstruct a finite group $G$ from its category of finite dimensional complex representations $\text{rep}_{\mathbb{C}}(G)$.

The answer depends on how we consider $\text{rep}_{\mathbb{C}}(G)$. Actually we cannot reconstruct $G$ if we only consider $\text{rep}_{\mathbb{C}}(G)$ as an abelian category or monoidal category, but we can reconstruct $G$ if we consider $\text{rep}_{\mathbb{C}}(G)$ as a symmetric monoidal category.

The proof outlined in the book is to consider the forgetful functor $$ F: \text{rep}_{\mathbb{C}}(G)\to \text{vect}_{\mathbb{C}} $$ to the symmetric monoidal category of finite dimensional $\mathbb{C}$-vector spaces. Then we show there is an isomorphism $$ G\to \text{Aut}^{\otimes}(F) $$ where $\text{Aut}^{\otimes}(F)$ denotes the group of natural isomorphisms of $F$ which are compatible with the tensor product on $\text{rep}_{\mathbb{C}}(G)$.

I want to see the impact $\text{Aut}^{\otimes}(F)$ when we add the symmetric structure on the monoidal categories. But it seems to me that there is no difference between the symmetric monoidal case and merely monoidal case. For example, symmetric means the following diagram commutes for $\alpha\in \text{Aut}^{\otimes}(F)$ $$ \require{AMScd} \begin{CD} F(V\otimes W)@>F(S_{V,W})>> F(W\otimes V)\\ @V \alpha_{V\otimes W} VV @VV \alpha_{W\otimes V} V\\ F(V\otimes W) @>>F(S_{W,V})> F(W\otimes V) \end{CD} $$ where $S_{V,W}$ is the symmetry functor in $\text{rep}_{\mathbb{C}}(G)$.

However since $F$ is the forgetful functor we automatically have $$ F(S_{V,W})\cong S_{F(V),F(W)} $$ and since $\alpha$ preserve the monoidal structure we have $$ \alpha_{V\otimes W}\cong \alpha_V\otimes \alpha_W. $$ Therefore since we have the commutative diagram $$ \begin{CD} F(V)\otimes F(W)@>S_{F(V),F(W)}>> F(W)\otimes F(V)\\ @V \alpha_{V} \otimes\alpha_{W} VV \circlearrowright @VV \alpha_{W}\otimes\alpha_{V} V\\ F(V)\otimes F(W) @>>S_{F(W),F(V)}> F(W)\otimes F(V) \end{CD} $$ the first diagram also commute. In conclusion there is no difference of $\text{Aut}^{\otimes}(F)$ whether we consider $\text{rep}_{\mathbb{C}}(G)$ as a symmetric monoidal category or just a monoidal category.

What did I do wrong in this argument?