One needs to be careful. One cannot recover the group $G$ from the tensor category alone, but only with the data of category, fiber functor. There are examples of non-isomorphic (finite, even) groups with equivalent categories of representations. For instance, see Pasquale Zito's answer to this question:

Finite groups with the same character table

However, as is discussed in the paper Zito links to, remembering the symmetry on the categories recovers the group, up to isomorphism. I'm not sure who it's due to.

One needs to be careful. One cannot recover the group $G$ from the tensor category alone, but only with the data of category, fiber functor. There are examples of non-isomorphic (finite, even) groups with equivalent categories of representations. For instance, see Pasquale Zito's answer to this question:

Finite groups with the same character table

However, as is discussed in the paper Zito links to, remembering the symmetry on the categories recovers the group, up to isomorphism. I'm not sure who it's due to.