A series of mononuclear hexacoordinate Co(ii) complexes with the 4-amino-3,5-bis(2-pyridyl)-1,2,4-triazole (abpt) ligand and various linear and non-linear pseudohalides, such as NCSe (selenocyanate), N{C(CN)} (1,1,3,3-tetracyano-2-azapropenide, tcap), NOC(CN) (nitrodicyanomethanide, nodcm), C(CN){C(CN)} (1,1,2,3,3-pentacyanopropenide, pcp), NONCN (nitrocyanamide, nca), and ONC(CN) (nitrosodicyanomethanide, ndcm), was prepared. X-ray analyses revealed the formation of the complexes with the general compositions [Co(abpt)(solv)]X (solv = HO and X = tcap (1), solv = HO and X = nodcm (2), solv = CHOH and X = pcp (3)) or [Co(abpt)(X)] (X = nca (4), X = NCSe (5), X = ndcm (6)). The impact of axial co-ligands (solv or X) on the magnetic properties was investigated experimentally by measuring temperature- and field-dependent static (DC) and dynamic magnetic (AC) data as well as theoretically using the CASSCF/NEVPT2, AILFT, and AOM methods. Large magnetic anisotropy was found for all complexes 1-6 and was treated either by the spin Hamiltonian or with the Hamiltonian including the orbital angular momentum. Furthermore, the AC susceptibility measurements confirmed the slow relaxation of the magnetization in a non-zero static magnetic field, thus these complexes can be classified as field-induced single-molecule magnets with an estimated energy barrier U up to 100 K.