Question:Characteristic subgroup other isomorphic subgroups

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Q: If H is a characteristic subgroup of G, does that mean there is no other subgroup of G isomorphic to H?

A: Not in general.

Call a subgroup of a group an isomorph-free subgroup if there is no other subgroup of the group isomorphic to it. Then, isomorph-free implies characteristic but characteristic not implies isomorph-free. In fact:

Basically, characteristic just means invariant under automorphisms of the whole group. Being the only subgroup of its type in any of the above senses is good enough to force characteristicity, but a subgroup could be characteristic for many other reasons.