Weak Cayley Table Groups

LetG, Hbe groups. We study bijectionsf:G → Hwhich preserve conjugacy classes and satisfy: for allg, g′ Gthe elementsf(gg′) andf(g)f(g′) are conjugate. These are related to 2-characteristics of groups, objects originally studied by Frobenius and more recently by K. W. Johnsonet. al.IfG = H, then any automorphism or anti-automorphism satisfies these conditions. We call thesetrivial.We show, among other things, that any suchf:G → GforGbeing the symmetric groupSn,n > 0, dihedral groupDn,n {∞}, or the free groupFn,n ≠ 3, is trivial.