Let $X = (V,E)$ be a connected trivalent graph, $e = (v_1,v_2)$ an edge of $X$. We define:
$A^{(0)}$ as the pointwise stabilizer in $Aut(X)$ of all vertices in $V_0=\{v_1, v_2\}$ where $Aut(X)$ is the automorphisms of $X$.
Then according to a theorem of Tutte, $Aut_e(X)$, the group of all those automorphisms of $X$ which stabilize the edge $e$, is a $2$-group. The proof states that, $A^{(0)}$ has index 1 or 2 in $Aut(X)$.
Index 1 means $A^{(0)}= Aut(X)$, it is understandable when $Aut_e(X)=Aut(X)$, but when index is 2?
How can we prove that?