  A minimal equicontinuous action by homeomorphisms of a discrete group
$\Gamma$ on a Cantor set $X$ is locally quasi-analytic, if each homeomorphism
has a unique extension from small open sets to open sets of uniform diameter on
$X$. A minimal action is stable, if the actions of $\Gamma$ and of the closure
of $\Gamma$ in the group of homeomorphisms of $X$, are both locally
quasi-analytic.
  When $\Gamma$ is virtually nilpotent, we say that $\Phi \colon \Gamma \times
\mathfrak{X} \to \mathfrak{X}$ is a nilpotent Cantor action. We show that a
nilpotent Cantor action with finite prime spectrum must be stable. We also
prove there exist uncountably many distinct Cantor actions of the Heisenberg
group, necessarily with infinite prime spectrum, which are not stable.
