The problem is :
Prove or disprove : If $g$ and $h$ are two elements of a finite group $G$, then there exists a group $H$, an extension of $G$ such that $g$ and $h$ are conjugates in $H$.
Actually, I found on Wikipedia that it can be done using the HNN extension and Britton's lemma, it is a corollary of Britton's lemma, but the proof is not provided there.
Can someone please help me with the proof?
Hint: conjugate elements have the same order. (Why?)