Let $H$ be a connected algebraic group over an algebraically closed field $k$, and $I$ a finite group which acts on $H$ through group scheme morphisms. Denote by $Rep(H)$ the category of finite dimensional algebraic representations of $H$ and $\omega:Rep(H)\rightarrow Vect_k$ the natural fiber functor. We can associate to each $i\in I$ an exact strict monoidal endofunctor $\rho_i$ of $Rep(H)$ such that $\omega\circ \rho_i=\omega$. In other words $I$ acts on $Rep(H)$. On the other hand we can form the subgroup of $I$ invariants $H^I\subset H$ which is a closed subgroup of $H$.
My question is the following: Is there some ambient 2-category $C$ of categories (e.g category of k-linear finitely cocomplete symmetric monoidal categories with finitely cocontinuous monoidal functors and tensor natural transformations between them) containing $Rep(H)$ and $Rep(H^I)$ such that $Rep(H^I)=Rep(H)/I$ in $C$ in the following sense:
The datum of a map $F:Rep(H)\rightarrow T$ in $C$, along with 2-isomorphisms $F\circ\rho_i\cong F$ for each $i\in I$ is equivalent to, the datum of a map (unique up to 2-isomorphism) $Rep(H^I)\rightarrow T$ making the following diagram commute up to 2-isomorphism:
\begin{array}{cc} Rep(H) & \xrightarrow{F} & T\\ Res_H^{H^I} \downarrow & \nearrow \\ Rep(H^I) & \end{array}.
Here is what I know:
If $C$ is the category of Tannakian categories over $Vect_k$ (with the fixed functor to $Vect_k$ being the fiber functor) with 1-morphisms exact monoidal functors, above statement is true. It can be proven by translating the situation to maps between associated groups.
For $C$ just the category of symmetric monoidal categories with symmetric monoidal functors between them, it is false.
I believe it might be true if $C$ is the category of k-linear finitely cocomplete symmetric monoidal categories with finitely cocontinuous monoidal functors. For instance if $H$ is a torus, the statement is true for this choice of $C$.
