Let $X$ be a trivalent graph. We denote by ${\rm Aut}_e(X)$ the subgroup of ${\rm Aut}(X)$ such as fix the edge $e$,
i.e., $\forall \phi \in{\rm Aut}_e(X)$ if $e = (v_1, v_2)$ then $ \phi(v_1) = v_2$ and $ \phi(v_2) = v_1$
or $( \phi(v_1) = v_1$ and $ \phi(v_2) = v_2$.
Here $X_r$ is the subgraph consisting of all vertices and all edges of the graph$X$ which appear in paths of length $\leq r$ through $e = (a, b)$.
There are natural homomorphisms:
$$\pi_r :{\rm Aut}_e(X_{r+1}) \rightarrow{\rm Aut}_e(X_r)$$ in which $\pi_r(\sigma)$ is the restriction of $\sigma$ to $X_r$.
Let $K_r$ is the kernel of $\pi_r$, in this paper, page 48, it says kernel $K_r$ fixes all elements (vertices) of $X_r$.
But since $\pi_r(\sigma)$ is the restriction of $\sigma$ to $X_r$, then the kernel, say $N$ should be all the permutations that fixes vertices of $X_{r+1} \setminus X_r$ instead of $X_r$, and ${\rm Aut}_e(X_r) = {\rm Aut}_e(X_{r+1})/N$?
What am I missing here? Please help, thank you.
It is intuitive to thing that any automorphism of $X_r$ can be found in automorphism of $X_{r+1}$, but in the case of kernel, this does not make thing consistent with the paper.
Since $\pi_r$ is a homomorphism of groups, its kernel $K_r$ is by definition the set $$K_r = \{ σ \in \mathrm{Aut}_e(X_{r+1}) :\pi_r(σ)=1\},$$ where $1$ denotes the identity automorphism of $X_r$. Since $1$ fixes all elements of $X_r$, and $\pi_r(σ) =σ\vert_{X_r}$, we must have that $σ$ fixes all elements of $X_r$. Conversely, any element that fixes all vertices must lie in the kernel.
You're right that it's intuitive to think that any automorphism of $X_r$ can be found in an automorphism of $X_{r + 1}$, but the important thing to note is that there might be multiple ways to extend an automorphism of $X_r$ to an automorphism of $X_{r + 1}$. That is, there might be multiple paths of length $r + 1$ that contain the same path of length $r$, and you could have an automorphism fixing these, or exchanging these.
The idea behind the kernel is that it captures the idea of when automorphisms are the same. Basically, if you look at automorphisms of $X_{r + 1}$ and you want to view them as automorphisms of $X_r$ then surely automorphisms that act the same should be identified. The kernel identifies everything that acts as the identity, and then the quotient $\mathrm{Aut}_e(X_{r + 1})/K_r$ consists of cosets that each act as different automorphisms on $X_r$.
If $\pi_r$ is surjective (which by your question I assume it is), then you can identify $\mathrm{Aut}_e(X_{r + 1})/K_r = \mathrm{Aut}_e(X_r)$.