The etale fundamental group is an inverse limit of automorphism groups of finite etale coverings. We can see the geometry of etale fundamental group very well from etale coverings just like topologically fundamental group. But the tannakian fundamental groups are defined as the automorphism of a fiber functor of a tensor category, this is definition abstract for me.

I want ask that how to see the geometry of tannakian fundamental groups?

What is the relation between first etale cohomology and pro-unipotent fundamental group?

I would like to extend my question. If we consider a hyperbolic curve $X$ over a local field $K$ with valuation ring $R$, there is an natural Galois action (i.e., outer Galois action) of $G_{K}$ on the etale fundamental group $\pi_{1}(X)$. From this outer Galois action, we can understand some geometry of $X$ and their reduction. For example, there is a good reduction criterion in terms of pro-$l$ fundamental groups (Oda, Tamagawa).

My question is: Dose there exist some similar Galois actions or criterions for Tannakian fundamental groups? or dose there exist some anabelian type theorems for Tannakian fundamental groups?