Let $0 \to A \to B \to C \to 0$ be an exact sequence of finite abelian groups. Assume that $B$ and $C$ is a square (i.e. there are groups $D,E$ such that $B \cong D^2$, $C \cong E^2$). Does this imply that also $A$ is a square?
Of course we may assume that $A,B,C$ are finite abelian $p$-groups for some prime $p$.