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Let $G$ be a finite abelian group which acts symplectically on a K3 surface. Then the N\'eron-Severi lattice of the general projective K3 surface admitting $G$ symplectic action is computed by Garbagnati and Sarti. In this paper we consider a special $4$-dimensional subfamily of the $7$-dimensional family of projective K3 surfaces with $\mathbb{Z}_2^2$ symplectic action. If $X$ is one of these special K3 surfaces, then it arises as the minimal resolution of a specific $\mathbb{Z}_2^3$-cover of $\mathbb{P}^2$ branched along six general lines. We show that the N\'eron-Severi lattice of $X$ is generated by an arrangement of $24$ smooth rational curves, and that $X$ specializes to the Kummer surface $\textrm{Km}(E_i\times E_i)$. We relate this $4$-dimensional family to other well known families of K3 surfaces, namely the minimal resolution of the $\mathbb{Z}_2$-cover of $\mathbb{P}^2$ branched along six general lines, and the corresponding Hirzebruch-Kummer covering of exponent $2$ of $\mathbb{P}^2$.

K3 surfaces with $\mathbb Z_2^2$ symplectic action