By the classical theory of Tannakian duality we know that every $k$-linear rigid abelian tensor category ($k$ a field) which has a fibre functor to $\mathrm{Vec}_k$ (finite dim. vector spaces over $k$) is equivalent to the category of representations of an affine group scheme over $k$ (or equivalently the category of comodules over a commutative Hopf algebra in $\mathrm{Vec}_k$ ).

I'm interested in the more general case where one has a "good" tensor category $C$ with a fibre functor to another "good" tensor category $D$ (instead of $\mathrm{Vec}_k$ ). What happens in this case? Is $C$ equivalent to the category of comodules over a Hopf algebra in $D$? If yes, how this Hopf algebra can be constructed?

As an example consider this two special cases:

1- $\mathrm{PHS}_\mathbb{Q} \to \mathrm{Vec}^\bullet_\mathbb{Q}$: weight grading functor,

2- $\mathrm{MHS}_\mathbb{Q} \to \mathrm{PHS}_\mathbb{Q}$: sum of graded quotients functor.