  Coboundary expansion (with $\mathbb{F}_2$ coefficients), and variations on
it, have been the focus of intensive research in the last two decades. It was
used to study random complexes, property testing, and above all Gromov's
topological overlapping property.
  In part I of this paper, we extended the notion of coboundary expansion (and
its variations) to cochains with permutation coefficients, equipped with the
normalized Hamming distance. We showed that this gives a unified language for
studying covering stability of complexes, as well as stability of group
homomorphisms -- a topic that drew a lot of attention in recent years.
  In this part, we extend the theory to the permutation coefficients setting.
This gives some new results, even for $\mathbb{F}_2$ coefficients, opens
several new directions of research, and suggests a pattern to proving the
existence of non-sofic groups.
  Along the way, we solve the dimension $2$ case of a problem of Gromov,
exhibiting a family of bounded degree coboundary expanders with $\mathbb{F}_2$
coefficients.
