Centerless group that is upward-closed normal in its automorphism group

Definition
A centerless group that is upward-closed normal in its automorphism group is defined as follows: it is a centerless group, and when it is embedded inside its automorphism group via the action by conjugation, every subgroup of the automorphism group containing it, is a normal subgroup of the automorphism group.

Stronger properties

 * Weaker than::Complete group
 * Weaker than::Centerless group that is maximal in its automorphism group
 * Weaker than::Centerless group in which every automorphism is normal-extensible: