Suppose we have a finite group $G$ and a function $f:G\to G$ that satisfies the following property for all $a,b\in G$: $$af(ab)b=bf(ba)a$$ Under what conditions (or, for what groups $G$) can such a function $f$ exist? When such a function exists, what can we say about how many such functions there are? Are there any other interesting properties of $f$ implied by the above functional equation?
After playing around with this for a while, I noticed that
- If $G$ is abelian, then all functions $f:G\to G$ trivially satisfy this property
- For all $a\in G$, $a^2$ commutes with $f(e)$
- For all $a\in G$, $a$ commutes with $f(a)$
- For all $z\in Z(G)$ (the center of $G$) and all $a\in G$, $a$ commutes with $f(az)$