Can we classify finite 2-generated groups $G$ satisfying the following property:


For any pair $x,y$ which generate $G$, the pair $x,yxy^{-1}$ also generates $G$.


By the comments, no nontrivial abelian group can satisfy this property, so I suppose the first question is: Do such groups $G$ exist?



Can we classify all finite 2-generated groups $G$ such that if $x,y$ generate $G$, then so does $x,yxy^{-1}$?
**Theorem** The only finite group satisfying the condition is the trivial group.


*Proof.* Let $G$ be a nontrivial finite group and $S$ be a simple quotient of $G$, which satisfies the condition by Gaschutz's lemma. Then $S$ is non-abelian as mentioned above. If $x\in S$ is any involution, then by well-known result of Guralnick and Kantor in [Probalistic generation of finite simple groups](https://doi.org/10.1006/jabr.2000.8357), there exists an element $y\in S$ such that $S=\langle x,y\rangle$ while $\langle x,yxy^{-1}\rangle$ is a dihedral group, that is, $S\neq\langle x,yxy^{-1}\rangle$, a contradiction.


