How hard is a variant of graph automorphism problem?
If I understood well the question, this is an attempt to prove that your problem is $GI$-complete.

Given two graphs $G_1, G_2$ connect all the nodes of $G_1$ to a new node $x$ and all the nodes of $G_2$ to a new node $y$. Then connect $x$ and $y$ with a path of new nodes: $ x_1, x_2, ... x_k,\; z\;, y_k, ...,y_2, y_1$ (there is an odd number of nodes between $x$ and $y$, and $z$ is the central node).



Now you can pick some pairs of nodes at the two endpoints: $(x,y),(x_1,y_1),(x_2,y_2),...$. In this way you end up with an instance of your problem that has a solution if and only if the two original graphs are isomorphic.