  We construct exactly solvable models for a wide class of symmetry enriched
topological (SET) phases. Our construction applies to 2D bosonic SET phases
with finite unitary onsite symmetry group $G$ and we conjecture that our models
realize every phase in this class that can be described by a commuting
projector Hamiltonian. Our models are designed so that they have a special
property: if we couple them to a dynamical lattice gauge field with gauge group
$G$, the resulting gauge theories are equivalent to modified string-net models.
This property is what allows us to analyze our models in generality. As an
example, we present a model for a phase with the same anyon excitations as the
toric code and with a $\mathbb{Z}_2$ symmetry which exchanges the $e$ and $m$
type anyons. We further illustrate our construction with a number of additional
examples.
