Consider the commutative diagram of finite abelian groups
$\require{AMScd}$
\begin{CD}
0@>>> A @>i>> B@>\pi>> C@>>> 0\\
\ @VV 0 V@VVfV@VV 0 V\\
0@>>>A @>>i> B@>>\pi> C@>>> 0
\end{CD}
where all maps are homomorphisms, the rows are exact, and the leftmost and the rightmost vertical map are zero? **Is the middle map $f$ also zero?**

It is clear that $f\circ f=0$ but this is all that seems to follow from lazy diagram chasing, hence I suspect $f$ need not be zero, in general. If so, what is a counterexample?