Is it true that for every finite group $G$ we can find an extension $K\geq G$ such that for every two isomorphic subgroups $H$ and $H'$ of $G$ and isomorphism $\varphi:H\rightarrow H'$ there exists an element $k\in K$ such that for every $h\in H$ we have $\varphi(h)=khk^{-1}$?
I just can't see any way of approaching this, nor any reason why this would or wouldn't hold, neither any example of such extension. I have almost finished the Algebra course on my university and I still have no idea how to approach this problem so I start to doubt in my knowledge. Do you have any advice?