Given a field $k$ let $\Omega(k)$ be the set of algebraic closures of $k$. 

$\Omega(k)$ is obviously a groupoid. At each element $\bar{k}$ of $\Omega(k)$ we have its automorphism group over $k$, which is a copy of $Gal(\bar{k}/k)$. Any two copies can be identified via isomorphisms of respective algebraic closures. 

Question 1: What is this structure "groupoid fibered in (automorphism) groups" called? Is there a reference for it?

Question 2: Can we see automorphisms of $Gal(\bar{k}/k)$ in a natural way in this set up, e.g., via some fiber functor? (In fact what are the automorphism groups of some well-known Galois groups, for $k = \mathbb{Q}$ or $\mathbb{F}_p$ for example?)