  Let $G$ be a finite abelian group which acts symplectically on a K3 surface.
The N\'eron-Severi lattice of the projective K3 surfaces admitting $G$
symplectic action and with minimal Picard number is computed by Garbagnati and
Sarti. We consider a $4$-dimensional family of projective K3 surfaces with
$\mathbb{Z}_2^2$ symplectic action which do not fall in the above cases. If $X$
is one of these 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 $X$ with minimal Picard
number is generated by $24$ smooth rational curves, and that $X$ specializes to
the Kummer surface $\textrm{Km}(E_i\times E_i)$. We relate $X$ to the K3
surfaces given by 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 $\mathbb{P}^2$.
