Group action on fibre functor (I asked this question on mathstack here: https://math.stackexchange.com/questions/4413271/group-action-on-fibre-functor. After getting no response and being suggested in the comments to post it here, I am posting this here.)
Let $C$ be a neutral Tannakian category (ie. it is rigid tensor Abelian category where hom sets are $k$-vector spaces and End$(1)=k$ and there is a fibre functor $w$ from $C$ to category of vector spaces such that $w$ is a exact faithful tensor functor.)
Let group $G$ act on $C$ in the sense that for each $g \in G$ we have a functor $a_g : C \to C$ such that $a_g a_h$ and $a_{gh}$ are naturally isomorphic. Further assume the group acts by tensor functors.

Simpson in his paper "Higgs Bundles and Local Systems" says that "by transport of structure, $G$ also acts on ${\rm End}(w,C)$ where the endomorphism algebra is just the endomorphism of the fibre functor."


My question is : How do we get such an action?

A typical element in ${\rm End}(w,C)$ is a tuple ${f_V}$ where $f_V \in{\rm End}(w(V))$ and makes required diagrams commute. (It's just the data of natural transformation nothing else).
 A: Deligne proved that assuming that $C$ is Tannakian and $K$-linear, where $K$ is an algebraically closed field of characteristic zero, then there is a unique fiber functor from C to $Vec_K$. If you have a symmetric monoidal functor $F:C\to D$, and you have forgetful functors $F_C:C\to Vec_K$ and $F_D:D\to Vec_K$ this implies in particular that the functors $F_DF$ and $F_C$ are equivalent, due to uniqueness.
Tannaka reconstruction tells you that $C\cong Rep-H_1$ and $D\cong Rep-H_2$, where $H_1$ and $H_2$ are some groups, reconstructed as $H_1=Aut(F_C)$ and $H_2=Aut(F_D)$. The equivalence $F_DF\cong F_C$ then gives you a homomorphism of groups $\phi:H_2\to H_1$, and it can be shown that $F$ is induced by the pullback along $\phi$. Moreover, the homomorphism $\phi$ is unique up to composition with conjugation in $H_1$.
In your case $C=D$, $H_1=H_2$ and $F=a_g$. So you get in fact a homomorphism of groups $G\to Out(H)$. There should be a reason why this is liftable to a proper action, not only an outer one, but I do not know the context well enough.
