As we know, the functional equation $f(xf(y))=f(f(x)y)$ was completely solved in abelian groups (by J. G. Dhombres, Solution... $f(x\ast f(y))=f(y\ast f(x))$, Aequationes Math. 15 (1977), 173--193, doi:10.1007/BF01835648).
But in the book "Functional Equations in Several Variables" (J. Aczel and J. Dhombres), they mentioned that:
it has been generalized for nonabelian groups but with additional assumptions concerning $f$, and the problem is open in general (also in abelian quasigroups).
Now, my question is:
(1) What is the additional assumptions (on $f$) in nonabelian groups?
(2) Any progress in abelian quasi-groups?
(3) Are there any papers or books regarding the problems?
