Supersingular reduction of Kummer surfaces in residue characteristic $2$

Given an abelian surface $A$, defined over a discrete valuation field and having good reduction, does the attached Kummer surface $\mathrm{K}\mathrm{m}(A)$ also have good reduction? In this paper we give an affirmative answer in the extreme case, that is, when the abelian surface has supersingular reduction in characteristic $2$.