The automorphism group of a complementary prism

Given a finite simple graph Γ on n vertices its complementary prism is the graph $\Gamma \overline{\Gamma }$ that is obtained from Γ and its complement $\overline{\Gamma }$ by adding a perfect matching where each its edge connects two copies of the same vertex in Γ and $\overline{\Gamma }$. It generalizes the Petersen graph, which is obtained if Γ is the pentagon. The automorphism group of $\Gamma \overline{\Gamma }$ is described for an arbitrary graph Γ. In particular, it is shown that the ratio between the cardinalities of the automorphism groups of $\Gamma \overline{\Gamma }$ and Γ can attain only the values 1, 2, 4, and 12. It is shown that $\Gamma \overline{\Gamma }$ is vertex-transitive if and only if Γ is vertex-transitive and self-complementary. Moreover, the complementary prism is not a Cayley graph whenever $n>1$.