Retraction-invariant characteristic subgroup

Definition
A subgroup of a group is termed a retraction-invariant characteristic subgroup if it satisfies the following two conditions:


 * 1) It is a retraction-invariant subgroup, i.e., any retraction of the whole group sends the subgroup to itself.
 * 2) It is a characteristic subgroup, i.e., any automorphism of the whole group sends the subgroup to itself.