Isomorph-automorphic subgroup

Symbol-free definition
A subgroup of a group is termed isomorph-automorphic if given any other isomorphic subgroup, there is an automorphism of the whole group, mapping the subgroup isomorphically to the other one.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed isomorph-automorphic if whenever there exists a subgroup $$K$$ of $$G$$ such that $$H$$ and $$K$$ are isomorphic groups, there exists an automorphism $$\sigma$$ of $$G$$ such that $$\sigma(H) = K$$.

Examples
In a finite cyclic group, every subgroup is isomorph-automorphic (in fact, every subgroup is isomorph-free: no two subgroups are isomorphic).

Similarly, in a finite elementary Abelian group, every subgroup is isomorph-automorphic. That's because given any two subspaces of a finite-dimensional vector space that have the same dimension, there is an automorphism of the whole space taking one to the other.

By Sylow's theorem, every Sylow subgroup is isomorph-automorphic.

On the other hand, many subgroups are not isomorph-automorphic:


 * 1) In $$\mathbb{Z}$$, the group of integers, any nontrivial subgroup is of the form $$m\mathbb{Z}$$, $$m \ne 0$$, hence is isomorphic to $$\mathbb{Z}$$. However, there is clearly no automorphism of $$\mathbb{Z}$$ mapping it to a proper subgroup.
 * 2) Any infinite-dimensional vector space is isomorphic to a subspace of codimension one, but there is no automorphism mapping the whole space to such a subspace.
 * 3) In the direct product $$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$$, the direct factor $$\mathbb{Z}/2\mathbb{Z}$$ is isomorphic to the subgroup of multiples of 2 in the direct factor $$\mathbb{Z}/4\mathbb{Z}$$. However, there is no automorphism taking the first to the second.
 * 4) In the dihedral group of order eight, the center is of order two, and any subgroup generated by a reflection has order two, so they are isomorphic. However, there is no automorphism taking the center to a subgroup generated by a reflection.

Stronger properties

 * Sylow subgroup
 * Order-unique subgroup
 * Order-automorphic subgroup
 * Order-conjugate subgroup
 * Isomorph-conjugate subgroup
 * Isomorph-free subgroup