  We study the phase retrieval property for orbits of general irreducible
representations of nilpotent groups, for the classes of simply connected
connected Lie groups, and for finite groups. We prove by induction that in the
Lie group case, all irreducible representations do phase retrieval.
  For the finite group case, we mostly focus on $p$-groups. Here our main
result states that every irreducible representation of an arbitrary $p$-group
with exponent $p$ and size $\le p^{2+p/2}$ does phase retrieval.
  Despite the fundamental differences between the two settings, our inductive
proof methods are remarkably similar.
