  We examine Hilbert-Schmidt stability (HS-stability) of discrete amenable
groups from several angles. We give a short, elementary proof that finitely
generated nilpotent groups are HS-stable. We investigate the permanence of
HS-stability under central extensions by showing HS-stability is preserved by
finite central quotients, but is not preserved in general. We give a
characterization of HS-stability for semidirect products $G\rtimes_\gamma
\mathbb{Z}$ with $G$ abelian. We use it to construct the first example of a
finitely generated amenable HS-stable group which is not permutation stable.
Finally, it is proved that for amenable groups flexible HS-stability is
equivalent to HS-stability, and very flexible HS stability is equivalent to
maximal almost periodicity. There is some overlap of our work with the very
recent and very nice preprint of Levit and Vigdorovich. We detail this overlap
in the introduction. Where our work overlaps it appears that we take different
approaches to the proofs and we feel the two works compliment each other.
