TITLE: Are automorphisms of extensions trivial?
QUESTION [5 upvotes]: Here is a statement for abelian categories which seems so basic I'm feeling embarrassed to have to ask whether it's true in general.  Suppose $0 \to A \to B \to C \to 0$ is an exact sequence with kernel $i$ and cokernel $\pi$.  Let $f$ be an automorphism of this extension: an isomorphism $f:B\to B$ such that $i = f\circ i$ and $\pi = \pi \circ f$.  Is $f$ necessarily the identity?  I feel sure that it must be, but I'm starting to worry since I'm having trouble proving it with diagram chasing.
Thanks!

REPLY [5 votes]: It is not true. Take $\mathbb{Z}_9$ with the generator $a$ and the automorphism $f:a\to 4a$.