Birational Rigidity and $\mathbb Q$-Factoriality of a Singular Double Cover of a Quadric Branched over a Divisor of Degree 4

We prove birational rigidity and calculate the group of birational automorphisms of a nodal $\mathbb Q$-factorial double cover $X$ of a smooth three-dimensional quadric branched over a quartic section. We also prove that $X$ is $\mathbb Q$-factorial provided that it has at most 11 singularities; moreover, we give an example of a non-$\mathbb Q$-factorial variety of this type with 12 simple double singularities.