Group in which every element is order-automorphic

Definition
A group in which every element is order-automorphic is a group such that, whenever two elements have the same order, there is an automorphism of the group sending the first element to the second element.

Stronger properties

 * Weaker than::Group whose automorphism group is transitive on non-identity elements
 * Weaker than::Group in which every element is order-conjugate

Weaker properties

 * Stronger than::Group in which any two elements generating the same cyclic subgroup are automorphic
 * Stronger than::Group in which every element is automorphic to its inverse