Finite direct power-closed characteristic subgroup

Definition
Suppose $$H$$ is a subgroup of a group $$G$$. We say that $$H$$ is finite direct power-closed characteristic in $$G$$ if the following holds for every natural number $$n$$: in the group $$G^n$$ which is defined as the external direct product of $$n$$ copies of $$G$$, the corresponding subgroup $$H^n$$ is a characteristic subgroup.

Extreme examples

 * The trivial subgroup in any group is a finite direct power-closed characteristic subgroup.
 * Every group is a finite direct power-closed characteristic subgroup of itself.

Examples of subgroup-defining functions

 * Center is finite direct power-closed characteristic: The center of a group always has this property. More generally, any bound-word subgroup, and in particular any marginal subgroup, is finite direct power-closed characteristic. Thus, all members of the finite part of the upper central series are finite direct power-closed characteristic.
 * The derived subgroup, and more generally any verbal subgroup or even any fully invariant subgroup, is finite direct power-closed characteristic. Hence, all members of the finite part of the lower central series as well as of the derived series are finite direct power-closed characteristic.