Given a $2n$ vertex undirected graph whose vertices are partitioned arbitrarily in pairs to say WLOG $(1,2)$, $(3,4)$, $\dots$, $(2n-1,2n)$. Call these vertices pairs as super vertices.
Call two such graphs $2n$ vertex labelled graphs $G$, $H$ whose adjacencies are $A$ and $B$ respectively $\mathsf{vertex}$-$\mathsf{pair}$-$\mathsf{isomorphic}$ iff there is a permutation $P$ such that $A=PBP'$ on the condition that the permutation $P$ permutes only supervertices (only a subset $n!$ of permutations of $(2n)!$ allowed). In particular, the permutation must map even-numbered vertices in $G$ to even-number vertices in $H$; and if vertex $2i$ in $G$ is mapped to vertex $2j$ in $H$ by the permutation, then vertex $2i-1$ in $G$ must be mapped to $2j-1$ in $H$.
Is $\mathsf{vertex}$-$\mathsf{pair}$-$\mathsf{isomorphic}$ $\mathsf{GI}$-$\mathsf{complete}$?