  We propose a family of order parameters to detect the symmetry
fractionalization class of anyons in 2D topological phases. This
fractionalization class accounts for the projective, as opposed to linear,
representations of the symmetry group on the anyons. We focus on quantum double
models on a lattice enriched with an internal symmetry in the framework of
$G$-isometric projected entangled pair states. Unlike previous schemes based on
reductions to effective 1D systems (dimensional compactification), the order
parameters presented here can be probed on genuinely two-dimensional
geometries, and are local: they rely on operations on few neighbouring
particles in the bulk. The power of these order parameters is illustrated with
several combinations of topological content and symmetry. We demonstrate that a
strictly finer phase distinction than that provided by dimensional
compactification can be obtained. As particular examples, the resolution power
of these order parameters is illustrated for a case with non-abelian
topological order, and for another with symmetries that involves permutation of
anyons.
