  Our paper deals with the investigation of extensions of commutative groups by
loops so that the quasigroups that result in the multiplication between cosets
of the kernel subgroup are T-quasigroups. We limit our study to extensions in
which the quasigroups determining the multiplication are linear functions
without constant term, called linear abelian extensions.
  We characterize constructively such extensions with left-, right-, or inverse
properties using a general construction according to an equivariant group
action principle. We show that the obtained constructions can be simplified for
ordered loops. Finally, we apply our characterization to determine the possible
cardinalities of the component loop of finite linear abelian extensions.
