Strong quasiautomorphism of a group

Definition
Let $$G$$ be a group. A bijective map $$f:G \to G$$ is termed a strong quasiautomorphism if both $$f$$ satisfies the following condition:


 * For any $$x,y \in G$$, we have that $$f(xy)f(y)^{-1}f(x)^{-1}$$ lies in the intersection of the normal closure of the subgroup generated by the commutator $$[x,y]$$.
 * For any $$x,y \in G$$, we have that $$f(xy)f(y)^{-1}f(x)^{-1}$$ lies in the intersection of the normal closure of the subgroup generated by $$f([x,y])$$.

Weaker properties

 * Stronger than::Quasiautomorphism of a group
 * Stronger than::1-automorphism of a group