An equivalent statement of split exactness

Question

Let $0\to A\to B\to C\to 0$ be a short exact sequence of finite abelian groups and $Hom(-,A)$ functor induces an exact sequence $0\to Hom(C,A)\to Hom(B,A)\to Hom(A,A)$. How to prove the equivalence bewteen the split exactness of the original sequence and the right exactness of the induced sequence?