On terraces for abelian groups

It is known that elementary abelian 2-groups of order at least 4 do not have terraces. Bailey's Conjecture is that these are the only groups which do not. We show that abelian groups, except possibly those of order coprime to 3 whose Sylow 2-subgroup is elementary abelian of order an odd power of two, satisfy Bailey's conjecture. A consequence of this is that all 2-nilpotent groups whose Sylow 2-subgroups are abelian, but not elementary abelian, have terraces.