Constructions of graphs with any possible two-fold automorphism and automorphism groups

It is known that the canonical double cover of any connected nonbipartite graph have an automorphism group of the form $H \rtimes \mathbb{Z}_2$, where $H$ is the set of automorphism which preserve bipartite parts. We construct connected nonbipartite and vertex determining graphs whose canonical double covers have auromorphisms group isomorphic to any semisimple product of $\mathbb{Z}_2$ with any abstract group $H$. Later we show, that the canonical double cover of any asymmetric graph have abelian automorphisms group of odd order. The above construction provides an example of asymmetric graph for any such group. By modifying the aforementioned construction we obtain graphs which have any possible number and type of graphs with isomorphic double covers.