Locally graded groups with all subgroups normal-by-finite

In a paper published in this journal [1], J. T. Buckley, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that is, groups G such that \H : Core(G)(H)\ is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that \H : Core(G)(H)\ less than or equal to n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BCF remains an open question. In this direction, the following problem is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.