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:

https://mathoverflow.net/questions/500/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.