Strong quasihomomorphism of groups

Definition
Suppose $$G,H$$ are groups. A function $$f:G \to H$$ is termed a strong quasihomomorphism of groups if $$f$$ sends the identity element of $$G$$ to the identity element of $$H$$, and further, for any $$x,y \in G$$, we have that $$f(xy)f(y)^{-1}f(x)^{-1}$$ is in the normal subgroup generated by $$f([x,y])$$.