Hom

If $G$ and $H$ are abelian groups, then the set $\text{Hom}(G, H)$ of all group homomorphisms from $G$ to $H$ is itself an abelian group: the sum $h+k$ of two homomorphisms is pointwise addition, that is, for all $u$ in $G$ , $(h+k)(u)=h(u)+k(u).$ A homomorphism of abelian groups $f:H_1\to H_2$ can induce a homomorphism $\bar f$ from $\text{Hom}(G,H_1)$ to $\text{Hom}(G,H_2)$ defined by $\bar f(h)=f\circ h.$ Which of the following statements is/are true?

Ⅰ. If $f$ is injective, then $\bar f$ is injective for all abelian group $G$ .

Ⅱ. If $f$ is surjective, then $\bar f$ is surjective for all abelian group $G$ .

Bonus: What about the converses of Ⅰ and Ⅱ?