Let $G$ be a bipartite graph, $L$ ($R$) be the set of vertices in the left (right) part. Consider a graph $T$ with the set of vertices $R \times L$ ( $L \times R$ ) in the left (right) part. For any $l_1,l_2 \in L$, $r_1, r_2 \in R$, vertices $(r_1,l_1)$ and $(l_2,r_2)$ are adjacent in $T$ iff $l_1$ and $r_2$ are adjacent in $G$ and $l_2$ and $r_1$ are adjacent in $G$. Then $T$ has an automorphism $F$: For any $l \in L$, $r \in R$, $F((r,l))=(l,r), F((l,r))=(r,l)$. It seems that if $T$ has any perfect matching, then $T$ has a perfect matching invariant under $F$. Any ideas on how to prove this?