Difference sets in abelian 2-groups

Examples of difference sets are given for large classes of abelian groups of order 22d + 2. This fills in the gap of knowledge between Turyn's exponent condition and Dillon's rank condition. Specifically, it is shown thatℤ/(2d)×ℤ/(2d + 2) andℤ/(2d + 1)×Z/(2d + 1) both admit difference sets, and these have many implications.