I have calculated the fundamental group of sphere with a diameter using Van-Kampen theorem, which is $Z$. So corresponding to subgroup $2Z$ there exist a two sheeted connected covering space. So existence is clear. But I am not getting idea how to find it explicitly. Any help in this direction shall be appreciated.
You can see that your space is homotopically equivalent to $S^1\vee S^2$ and the double cover space of $S^1\vee S^2$ is $S^2\vee S^1\vee S^2$.
This is not the double cover of your space but it is homotopically equivalent to that but it is easier to visualize. You can analogously construct the cover space of the sphere with diameter: you can see it as two copies of $S^2$ with say the corresponding north poles and south poles connected by an edge.