During my ongoing research project, I encountered the following question. If this statement is true, I believe it must be documented somewhere.
Let $G$ be a group such that every element commutes with all of its conjugates. Is it true that $ Z(G) \leq G' $ (where $G^\prime=[G,G]$, i.e., that $G$ is 2-nilpotent)?
So far, I have noted that the property regarding conjugacy classes leads to the identity $[x, y] = [y^{-1}, x]$, and that there are no counterexamples with size less than 400.
A group $G$ is $2$-Engel if and only if for every $x$ and $y$, $x$ commutes with $[x,y]=x^{-1}y^{-1}xy$. This is equivalent to each element $x$ commuting with all its conjugates.
If a $2$-Engel group has no elements of order $3$, then it is nilpotent of class $2$. But the free Burnside group of exponent $3$ and rank $3$, which has order $3^7$, is $2$-Engel but not $2$-nilpotent. I believe it is the smallest example.
In Groups in which the commutator subgroup satisfies certain algebraic conditions, J. Indian Math. Soc. (N.S.) 6 (1942) pp. 87-97, F. W. Levi showed that if $[[x,y],z]=[x,[y,z]]$ whenever two of $x$, $y$, and $z$ are equal, then $[[G,G],G]\subseteq Z(G)$ and its elements are of exponent $3$.
Note that the written condition is in fact equivalent to the $2$-Engel condition: if the condition holds and $x=y$, and similarly if $z=y$, then we have $[x,[x,z]]=[[x,x],z]=1$. For $x=z$, we have $[[z,y],z]=[z,[y,z]] = [[y,z],z]^{-1} = [y,[z,z]]^{-1}=1$.
Conversely, if $G$ satisfies the $2$-Engel condition, then it satisfies that $[[x,y],z]=[x,[y,z]]$ whenever two of $x$, $y$, and $z$ are equal. If $x=y$, then $[y,[y,z]]=1 = [[y,y],z]$. Symmetrically if $z=y$. And if $x=z$, then $[[z,y],z]=1=[z,[y,z]]$.
A verification that the free Burnside group of exponent $3$ and rank $3$ is of class exactly three can be obtained from the normal form for free Burnside groups of exponent $3$ in Marshall Hall's The Theory of Groups, Section 18.2.
