On groups with all proper subgroups finite-by-abelian-by-finite

We show that if all proper subgroups of a locally graded group G are finite-by-abelian-by-finite, then G contains a finite normal subgroup N such that all proper subgroups of G/N are abelian-by-finite. Then we apply this result to the study of groups which are minimal-non-P also for related group properties P. Finally we see how for locally (solubleby-finite) groups of infinite rank, it is enough to restrict attention to the proper subgroups with infinite rank.