Subgroups of locally finite products of locally nilpotent groups

Let the locally finite group G be the product of two locally nilpotent subgroups A and B, and assume that H is a subgroup of G belonging to a group class F. The question is considered whether there exists a subgroup X of G containing H which belongs to F and satisfies X = (X boolean AND A)(X boolean AND B). Under various assumptions on G and F, necessary and sufficient conditions for the existence of such a subgroup X are obtained.