Abstract
We consider the isomorphism problem for quasigroups isotopic to a group and show that the isomorphism classes correspond to orbits of an action by the holomorph of the group upon pairs of permutations of the underlying set, with one of them fixing the neutral element. The results are applied to central quasigroups and to quasigroups of order 4.
We also characterize equationally those isotopes of a group in which one of the permutations is a group automorphism.