  We prove that every uniform approximate homomorphism from a discrete amenable
group into a symmetric group is uniformly close to a homomorphism into a
slightly larger symmetric group. That is, amenable groups are uniformly
flexibly stable in permutations. This answers affirmatively a question of Kun
and Thom and a slight variation of a question of Lubotzky. We also give a
negative answer to Lubotzky's original question by showing that the group
$\mathbb{Z}$ is not uniformly strictly stable. Furthermore, we show that
$\text{SL}_{r}(\mathbb{Z})$, $r\geq3$, is uniformly flexibly stable, but the
free group $F_{r}$, $r\geq 2$, is not. We define and investigate a
probabilistic variant of uniform stability that has an application to property
testing.
