Inverse property of nonassociative abelian extensions

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