Let $G$ be a group and $H$ is a subgroup of $G$, then there exists a normal subgroup $K$ of $G$, such that $H$ is isomorphic to $K$. Under such conditions, can we determine the structure of $G$ ?
This question comes from the discussion of Dedekind group in group theory, where a Dedekind group means that all of its subgroups are normal, in this case, we know that a Dedekind group is an Abelian
group or a direct product of Quaternion group $Q_8$ and an Abelian group $A$, where $A$ has no elements with order $4$.
So my problem can be regarded as a promotion of Dedekind group, it only requires isomorphic, I don’t know how to deal with this case.