Residually finite amenable groups that are not Hilbert–Schmidt stable

We construct the first examples of residually finite amenable groups that are not Hilbert–Schmidt (HS) stable. We construct finitely generated, class 3 nilpotent-by-cyclic examples and solvable linear finitely presented examples. This also provides the first examples of amenable groups that are very flexibly HS-stable but not flexibly HS-stable and the first examples of residually finite amenable groups that are not locally HS-stable. Along the way we exhibit (necessarily not-finitely-generated) class 2 nilpotent groups G = A\rtimes \Z with A abelian such that the periodic points of the dual action are dense, but it does not admit dense periodic measures. Finally, we use the Tikuisis–White–Winter theorem to show that all of the examples are not even operator-HS-stable; they admit operator norm almost homomorphisms that cannot be HS-perturbed to true homomorphisms.