TITLE: Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?
QUESTION [11 upvotes]: Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?

REPLY [6 votes]: I thought it worth adding a reference to this:
Miyata, Takehiko, Note on direct summands of modules, J. Math. Kyoto Univ. 7 1967 65–69. 
In the paper, the question of the OP is attributed to Matsumura and the solution to Toda.  Then a short argument is given for this case.
The author generalizes this result to the following (which is a quote from MathSciNet):

"Let $R$ be a commutative, noetherian ring and let $A$ be an
  $R$-algebra of finite type. Moreover, let $M$ be a finitely generated
  $A$-module and let $N$ be a submodule of $M$. Using the usual tools of
  homological algebra and noetherian ring theory, the author establishes
  the following pair of results. 
Theorem 1: If $M$ is isomorphic to $N\oplus M/N$, then $N$ is a  direct summand of $M$. 
Theorem 2: If $0\to N\otimes_R T\to M\otimes_R T$ is exact for all $A$-modules $T$ (i.e., $N$ is pure), then $N$ is a direct summand of
  $M$."