Irreducible symplectic 4-folds numerically equivalent to (K3)^{[2]}

First steps towards a classification of irreducible symplectic $4$-folds whose integral $2$-cohomology with $4$-tuple cup product is isomorphic to that of $(K3)^{[2]}$. We prove that any such $4$-fold deforms to an irreducible symplectic $4$-fold of Type A or Type B. A $4$-fold of Type A is a double cover of a (singular) sextic hypersurface and a $4$-fold of Type B is birational to a hypersurface of degree at most $12$. We conjecture that Type B $4$-folds do not exist.