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!
It is not true. Take $\mathbb{Z}_9$ with the generator $a$ and the automorphism $f:a\to 4a$.