  The study of extensions realizing affine datum is specialized to central
extensions in varieties with a difference term which leads to generalizations
of several classical theorems on central extensions from group theory. We
establish a 1-dimensional Hochschild-Serre sequence for central extension and
affine datum. This is used to develop a Schur-formula which characterizes the
$2^{\mathrm{nd}}$-cohomology group of regular datum in terms of the
transgression map and commutators in free presentations. We prove sufficient
conditions for the existence of covers and provide a cohomological
characterization of perfect algebras. The class of varieties with a difference
term contain all varieties of algebras with modular congruence lattices and so
the present work generalizes the classical results for groups and many
analogous results recently established for algebras of Loday-type which can be
recovered by specialization.
