  We show that amenability, the Haagerup property, the Kazhdan's property (T)
and exactness are preserved under taking second nilpotent product of groups. We
also define the restricted second nilpotent wreath product of groups, this is a
semi-direct product akin to the restricted wreath product but constructed from
the second nilpotent product. We then show that if two discrete groups have the
Haagerup property, the restricted second nilpotent wreath product of them also
has the Haagerup property. We finally show that if a discrete group is abelian,
then the restricted second nilpotent wreath product constructed from it is
unitarizable if and only if the acting group is amenable.
