Let $A=\mathbb{C}^2/\Lambda^2$, where $\Lambda=\mathbb{Z}+i\mathbb{Z}$, be an abelian surface. Then every body knows that the resolution of the quotient $A/<\pm>$ is a K3 surface.

Question: Is there an easy involution on the resulting K3 surface (may be induced from some action on A) such that the quotient is Enrique surface, i.e. is this K3 surface double cover of some Enrique surface?

I am not sure whether this is some thing easy or well-known.