  A result of Fiz Pontiveros shows that if $A$ is a random subset of
$\mathbb{Z}_N$ where each element is chosen independently with probability
$N^{-1/2+o(1)}$, then with high probability every Freiman homomorphism defined
on $A$ can be extended to a Freiman homomorphism on the whole of
$\mathbb{Z}_N$. In this paper we improve the bound to $CN^{-2/3}(\log
N)^{1/3}$, which is best possible up to the constant factor.
