Let $G$ be a finite group, and let $g,h\in G$ be two elements of the same order. Does there exist a larger finite group $G'\ge G$ such that $g,h$ are conjugate in $G'$?


If not, what is known about groups which have this property?



Given two elements $g,h$ in a finite group $G$ of the same order, does there exist a finite group $G'$ containing $G$ where they are conjugate?
[Two subgroups are isomorphic iff they are conjugate in some larger finite group.](https://groupprops.subwiki.org/wiki/Isomorphic_iff_potentially_conjugate_in_finite)


If you have two subgroups $A,B$ of a finite group $G$ and $\sigma: A\to B$ is an isomorphism. Then there exists a group $G'$ containing $G$ and an element $g'\in G'$ such that $\sigma$ is induced by conjugation by $g'$. For your question this follows by taking $A,B$ to be the cyclic groups generated by your two elements of the same order.


