Group in which every 1-automorphism is automorphism class-preserving

Definition
A group in which every 1-automorphism is automorphism class-preserving is a group with the property that any defining ingredient::1-automorphism of the group (i.e., any bijection that restricts to isomorphisms on cyclic subgroups) sends every element to an element that is in the same orbit under the action of the automorphism group, i.e., every element is sent within its automorphism class.