Biregular and birational geometry of quartic double solids with 15 nodes

Three-dimensional del Pezzo varieties of degree $2$ are double covers of $\mathbb{P}^{3}$ branched in a quartic. We prove that if a del Pezzo variety of degree $2$ has exactly $15$ nodes, then the corresponding quartic is a hyperplane section of the Igusa quartic or, equivalently, all such del Pezzo varieties are members of a particular linear system on the Coble fourfold. Their automorphism groups are induced from the automorphism group of the Coble fourfold. We also classify all birationally rigid varieties of this type.