Let $G$ be a finite group. If $a = bab$, is it true that $b^{2} = e$

Question

Let $G$ be a finite group and let $a,b \in G$. If $a = bab$, is it true that $b^{2} = e$. If not, find a counterexample.

It is clear that if $a = bab$ and $b^{2} = e$ are both true, then $ab = ba$. However, there exist groups (namely non-Abelian ones) with elements such that $ab \neq ba$. However, I am having trouble finding a non-Abelian group with elements such that $a = bab$, but $ab \neq ba$. How does one solve this problem?

Answer 1

No. For example, in $Q_8$ we have $jij=jk=i$.

Answer 2

Another counterexample is in $D_3$ where we have $f = R_{120}fR_{120}$.