Semihomomorphism of groups

Definition
Let $$G$$ and $$H$$ be groups. A semihomomorphism from $$G$$ to $$H$$ is a map $$f:G \to H$$ that satisfies, for all $$a,b \in G$$:

$$f(aba) = f(a)f(b)f(a)$$

Stronger notions

 * Antihomomorphism of groups
 * Homomorphism of groups

Weaker notions

 * 1-homomorphism of groups