The virtual braid group VBn, the virtual twin group VTn and the virtual triplet group VLn are extensions of the symmetric group Sn, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group Sn are the pure virtual braid group VPn, the pure virtual twin group PVTn and the pure virtual triplet group PVLn, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group PVLn, the commutator subgroup VTn′ of VTn and the commutator subgroup VLn′ of VLn. Our results complete the understanding of these groups, except that of VBn′, for which the existence of a finite presentations is not known for n≥4. We also prove that VLn/PVLn′ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it.