First note that I am naive to this research area.
Let $G$ be a finite group and so is $H$. Let $K$ be a subgroup of $G$ and $K'$ be a subgroup of $H$ and both subgroups are non-trivial. I know a map $\phi$ which is an isomorphism from $K$ to $K'$,
Question : Is it possible to extend this map $\phi$ from $G$ to $H$ ( assuming $G \cong H$)? if yes how?
Please feel free to share wiki-article or any thing on it, Thanks in advance.
No, this is not always possible.
For an example, take the dihedral group $D_4$ of order $8$. This group has a center of order $2$, and also a non-central element of order $2$. There is of course an isomorphism between the center and the subgroup generated by this element, but this cannot extend to an isomorphism from the group to itself, since such an isomorphism would preserve the center.