Take the tour ×  I have a question related to Mackey theory
applied to discrete nilpotent groups which are not torsion-free
and are infinite.  Let us suppose that $G$ is a type I infinite
discrete nilpotent group with normal abelian subgroup $N$ which
is regularly embedded. Also assume that the commutator subgroup
of $G$ is finite, and that $N$ is two-step. In order to describe
the unitary dual of $G$ using the Mackey machine, let us suppose
that we fix a character in the unitary dual of the normal
subgroup $N$. Let us also assume that the Little subgroup fixing
the given character is finite.    1. Do we know that the given
character of $N$ must extend to a representation of the
stabilizer subgroup of the fixed character as opposed to a
multiplier representation?   2. If we can extend the character
to a representation (which is not a multiplier representation)
the extended representation to the stabilizer subgroup inside
$G$ is unique up to multiplication by a constant. But how do we
construct the extended re