  We prove that every element of order 2 in the Brauer group of a complex
Kummer surface X descends to an Enriques quotient of X. In 'generic' cases this
gives a bijection between the set Enr(X) of Enriques quotients of X up to
isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces
of Picard rank 20 we prove that the fibres of the map from Enr(X) to Br(X)[2]
above the non-zero points have the same order.
