Free object actions give orbit coverings

Representatives and composition. If an arrow orbit has source $[x]$, translate any representative so its source is $x$. Two representatives in that orbit with source $x$ differ by an element fixing $x$, hence are equal by freeness. For composable arrow orbits, choose a representative $a:x\to y$ of the first and the unique representative $b:y\to z$ of the second with source $y$; define their product to be $[ab]$. Replacing $a$ by $na$ replaces $b$ by $nb$, so the product orbit is unchanged. Sources and targets are well defined because automorphisms preserve them.

Groupoid and covering. For three composable orbits, choose consecutive representatives $a,b,c$. Both parenthesizations have orbit $[abc]$, by uniqueness in 1.1 and associativity in $G$. Identity and inverse orbits satisfy the groupoid laws by the same construction. Thus the orbit maps define a functor $p$. It is bijective on every source-star by 1.1. The quotient is nonempty and connected because an arrow between representatives of any two object orbits exists in $G$. Consequently $p$ is a covering of connected groupoids.

Deck transformations. Each $n\in N$ acts as an automorphism over $G\mathrm{/}N$, so gives a deck transformation; these act transitively on every object fibre, which is an $N$-orbit. If a deck transformation fixes one object $x$, then it fixes every arrow with source $x$ by source-star injectivity, and hence fixes their targets. Connectedness reaches every object this way; source-star injectivity then fixes every arrow. For any deck transformation $d$, its value at $x$ is $nx$ for a unique $n$ by freeness. The deck transformation $n^{-1}d$ fixes $x$, so is the identity. Thus $N$ is exactly the deck group, with multiplication preserved by the left-action law, and the covering is normal.

Vertex sequence. For a loop orbit at $[x]$, its unique representative $a$ from $x$ ends at a unique $nx$, defining $\delta ([a])=n$. If $b:x\to mx$ represents a second loop orbit, the consecutive representative after $a$ is $nb:nx\to nmx$; hence $\delta ([a][b])=nm$, so $\delta$ is a homomorphism. Connectedness supplies an arrow $x\to nx$ for each $n$, proving surjectivity. Its kernel consists exactly of orbit classes represented by arrows $x\to x$, which are the image of $G_x$. The restriction of $p$ to $G_x$ is injective by source-star uniqueness. This proves the exact sequence and all assertions.