A subgroup M of an infinite group G is said to be nearly maximal if it is a maximal element of the set of all subgroups of G having infinite index; i.e. if the index |G:M| is infinite but every subgroup of G properly containing M has finite index in G. The near Frattini subgroup ψ(G) of an infinite group G can now be defined as the intersection of all nearly maximal subgroups of G, with the stipulation that ψ(G)=G if G has no nearly maximal subgroups. These concepts have been introduced by Riles . It was later proved by Lennox and Robinson  that a finitely generated soluble-by-finite group G is infinite-by-nilpotent if and only if all its nearly maximal subgroups are normal. It follows that in the class of finitely generated soluble-by-finite groups the property of being finite-by-nilpotent is inherited from the near Frattini factor group G/ψ(G) to the group G itself. In the study of ordinary Frattini properties of infinite groups, some analogies exist between the behaviour of finitely generated soluble groups and soluble minimax residually finite groups (see for instance  and ). This fact could suggest that a result corresponding to that of Lennox and Robinson also holds for soluble residually finite minimax groups. Unfortunately in this case the property of being finite-by-nilpotent cannot be detected from the behaviour of nearly maximal subgroups, this phenomenon depending on the fact that infinite soluble residually finite minimax groups may be poor of such subgroups.