Let G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K × B, with B a quasi-injective abelian group of odd order and either K = Q8 (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

In this paper a necessary and sufficient condition will be given for groups to be ν-isologic, with respect to a given variety of groups ν. Its is also shown that every ν-isologism family of a group contains a ν-Hopfian group. Finally we show that if G is in the variety ν, then every ν-covering group of G is a Hopfian group.