  We introduce an inseparable version of Kummer surfaces. It is defined as a
supersingular K3 surface in characteristic 2 with 16 smooth rational curves
forming a certain configuration and satisfying a suitable divisibility
condition. The main result is that such a surface admits an inseparable double
covering by a non-normal surface $A$ that is similar to abelian surfaces in two
aspects: its numerical invariants are the same as abelian surfaces, and its
smooth locus admits an abelian group structure.
