Hereditarily characteristic subgroup

Definition
A subgroup of a group is termed a hereditarily characteristic subgroup if every subgroup of it is a characteristic subgroup in the whole group.

Left realization
For a group $$G$$ to be realizable as a hereditarily characteristic subgroup in a bigger group $$K$$, a necessary condition is that $$G$$ be a Dedekind group, i.e., every subgroup of $$G$$ must be normal in $$G$$.