  We investigate fractionalization of non-invertible symmetry in (2+1)D
topological orders. We focus on coset non-invertible symmetries obtained by
gauging non-normal subgroups of invertible $0$-form symmetries. These
symmetries can arise as global symmetries in quantum spin liquids, given by the
quotient of the projective symmetry group by a non-normal subgroup as invariant
gauge group. We point out that such coset non-invertible symmetries in
topological orders can exhibit symmetry fractionalization: each anyon can carry
a "fractional charge" under the coset non-invertible symmetry given by a gauge
invariant superposition of fractional quantum numbers. We present various
examples using field theories and quantum double lattice models, such as
fractional quantum Hall systems with charge conjugation symmetry gauged and
finite group gauge theory from gauging a non-normal subgroup. They include
symmetry enriched $S_3$ and $O(2)$ gauge theories. We show that such systems
have a fractionalized continuous non-invertible coset symmetry and a
well-defined electric Hall conductance. The coset symmetry enforces a gapless
edge state if the boundary preserves the continuous non-invertible symmetry. We
propose a general approach for constructing coset symmetry defects using a
"sandwich" construction: non-invertible symmetry defects can generally be
constructed from an invertible defect sandwiched by condensation defects. The
anomaly free condition for finite coset symmetry is also identified.
