Finite direct power-closed characteristic not implies fully invariant

Verbal statement
It is possible to have a finite direct power-closed characteristic subgroup of a group that is not a fully invariant subgroup.

Facts used

 * 1) uses::Center is finite direct power-closed characteristic
 * 2) uses::Center not is fully invariant

Proof
The proof follows by piecing together facts (1) and (2).

An explicit example of (2), and hence of this result as well, is when the whole group $$G$$ is the direct product of S3 and Z2 and $$H$$ is the center of $$G$$.