On page 1 of Lecture 2, Algebra and Computation , (Course Instructor: V. Arvind), there is a theorem-
Theorem 2. With Graph − Iso (graph isomorphism) as an oracle, there is a polynomial time algorithm for Graph − Aut (automorphism) and vice-versa.
on page 2, the proof is given-
Proof. First we shall show that we can solve Graph−Iso with Graph−Aut as an oracle. We are given two graphs $G_1$ and $G_2$ and we need to create a graph G using the two such that the generating set of the automorphism should tell us if they are isomorphic or not. Let $G = G_1 \cup G_2$. Suppose additionally we knew that $G_1$ and $G_2$ are connected, then a single oracle query would be sufficient. If any of the generators of $Aut(G)$ interchanged a vertex in $G_1$ with one in $G_2$, then connnectivity should force $G_1 \simeq G_2$. But what if they are not connected? We then have this very neat trick, $G_1 \simeq G_2 \Leftrightarrow \bar G_1\simeq \bar G_2$ , and either $G_1$ or $\bar G_1$ has to be connected and hence one can check for connectivity and then ask the appropriate query.
$\bar G_1$ is the complement graph of $G_1$, I assume.
It is said that-
Suppose additionally we knew that $G_1$ and $G_2$ are connected, then a single oracle query would be sufficient.
But when we test graph isomorphism, $G_1 , G_2$ are two different disconnected graph. Is it why it went on like-
But what if they are not connected? We then have this very neat trick, $G_1 \simeq G_2 \Leftrightarrow \bar G_1\simeq \bar G_2$, and either $G_1$ or $\bar G_1$ has to be connected and hence one can check for connectivity and then ask the appropriate query.
So, are we checking whether there is an automorphism of $G' = \bar G_1 \cup \bar G_2$ that swaps a vertex of $\bar G_1$ with a vertex of $\bar G_2$?
How can we tell that this kind of swapping is an automorphism?
Yes, you are checking whether there is an automorphism of $G^{\prime} = \bar{G}_{1} \cup \bar{G}_{2}$ that swaps a vertex of $\bar{G}_{1}$ with a vertex of $\bar{G}_{2}$. You are using Graph-Aut as an oracle, so what you will do is use Graph-Aut to find the automorphism group of $G^{\prime}$ (returned as a set of generators), then just look at whether any of these generators swap vertices of $G_{1}$ with those of $G_{2}$. You don't need to do anything to tell if this swapping is an automorphism, it is automatically an automorphism because it's returned by Graph-Aut.