  Symmetry protected topological phases exhibit nontrivial short-ranged
entanglement protected by symmetry and cannot be adiabatically connected to
trivial product states while preserving the symmetry. In contrast, intrinsic
topological phases do not need ordinary symmetry to stabilize them and their
ground states exhibit long-range entanglement. It is known that for a given
symmetry group $G$, the 2D SPT phase protected by $G$ is dual to the 2D
topological phase exemplified by the twisted quantum double model
$D^{\omega}(G)$ via gauging the global symmetry $G$. Recently it was realized
that such a general gauging map can be implemented by some local unitaries and
local measurements when $G$ is a finite, solvable group. Here, we review the
general approach to gauging a $G$-SPT starting from a fixed-point ground-state
wave function and applying a $N$-step gauging procedure. We provide an in-depth
analysis of the intermediate states emerging during the N-step gauging and
provide tools to measure and identify the emerging symmetry-enriched
topological order of these states. We construct the generic lattice parent
Hamiltonians for these intermediate states, and show that they form an
entangled superposition of a twisted quantum double with an SPT ordered state.
Notably, we show that they can be connected to the TQD through a finite-depth,
local quantum circuit which does not respect the global symmetry of the SET
order. We introduce the so-called symmetry branch line operators and show that
they can be used to extract the symmetry fractionalization classes and symmetry
defectification classes of the SET phases with the input data $G$ and
$[\omega]\in H^3(G,U(1))$ of the pre-gauged SPT ordered state. We illustrate
the procedure of preparing and characterizing the emerging SET ordered states
for some Abelian and non-Abelian examples such as dihedral groups $D_n$ and the
quaternion group $Q_8$.
