Homomorphisms of group $\mathbb{Z}$ to itself send even integers to even integers

Question

Since any homomorphism $f$ of additive group $\mathbb{Z}$ to $\mathbb{Z}$ is of form $f(x)=a*x$, for some integer $a$, we conclude that $f(2x)=2ax \in 2\mathbb{Z}$, which means $f$ sends even integers to even integers. Am I right? Thanks in advance.