I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a finite tensor category together with a monoidal functor to vector spaces. See Theorem 5.3.12 of the Tensor Categories. This reconstruction is compatible with the one for groups, in the sense that the Hopf algebra you recover is $k[G]$. However, for the Hopf algebra one you look at tensor endomorphisms instead of tensor isomorphisms. If you start with H-mod for a Hopf algebra H and look at $\mathrm{Aut}^\otimes(F)$ you'll end up with the group G of grouplike elements of H, and Rep(G) will not be the same as H-mod because H is not spanned by its grouplike elements.

I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a finite tensor category together with a monoidal functor to vector spaces. See Theorem 5.3.12 of the Tensor Categories. This reconstruction is compatible with the one for groups, in the sense that the Hopf algebra you recover is $k[G]$. However, for the Hopf algebra one you look at all endomorphisms instead of tensor isomorphisms. If you start with H-mod for a Hopf algebra H and look at $\mathrm{Aut}^\otimes(F)$ you'll end up with the group G of grouplike elements of H, and Rep(G) will not be the same as H-mod because H is not spanned by its grouplike elements.

I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a finite tensor category together with a monoidal functor to vector spaces. See Theorem 5.3.12 of the Tensor Categories. This reconstruction is compatible with the one for groups, in the sense that the Hopf algebra you recover is $k[G]$. However, for the Hopf algebra one you look at tensor endomorphisms instead of tensor isomorphisms. If you start with H-mod for a Hopf algebra H and look at $\mathrm{Aut}^\otimes(F)$ you'll end up with the group G of grouplike elements of H, and Rep(G) will not be the same as Rep(H).

I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a finite tensor category together with a monoidal functor to vector spaces. See Theorem 5.3.12 of the Tensor Categories. This reconstruction is compatible with the one for groups, in the sense that the Hopf algebra you recover is $k[G]$. However, for the Hopf algebra one you look at tensor endomorphisms instead of tensor isomorphisms. If you start with H-mod for a Hopf algebra H and look at $\mathrm{Aut}^\otimes(F)$ you'll end up with the group G of grouplike elements of H, and Rep(G) will not be the same as H-mod because H is not spanned by its grouplike elements.