  Symmetry fractionalization (SF) on topological excitations is one of the most
remarkable quantum phenomena in topological orders with symmetry, i.e.,
symmetry-enriched topological phases. While much progress has been
theoretically and experimentally made in 2D, the understanding on SF in 3D is
far from complete. A long-standing challenge is to understand SF on looplike
topological excitations which are spatially extended objects. In this work, we
construct a powerful topological-field-theoretical framework approach for 3D
topological orders, which leads to a systematic characterization and
classification of SF. For systems with Abelian gauge groups ($G_g$) and Abelian
symmetry groups ($G_s$), we successfully establish equivalence classes that
lead to a finite number of patterns of SF, although there are notoriously
infinite number of Lagrangian-descriptions of the system. We compute
topologically distinct types of fractional symmetry charges carried by
particles. Then, for each type, we compute topologically distinct statistical
phases of braiding processes among loop excitations and external symmetry
fluxes. As a result, we are able to unambiguously list all physical observables
for each pattern of SF. We present detailed calculations on many concrete
examples. As an example, we find that the SF in an untwisted
$\mathbb{Z}_2\times \mathbb{Z}_2$ topological order with $\mathbb{Z}_2$
symmetry is classified by $ (\mathbb{Z}_2)^6\oplus (\mathbb{Z}_2)^2\oplus
(\mathbb{Z}_2)^2\oplus (\mathbb{Z}_2)^2$. If the topological order is twisted,
the classification reduces to $(\mathbb{Z}_2)^6$ in which particle excitations
always carry integer charge. Pure algebraic formalism of the classification is
given by: $ \bigoplus_{\nu_i} \mathcal H^4 ( G_g\leftthreetimes_{\nu_i}G_s, U
(1)) /\Gamma_\omega ({\nu_i}) \,$. Several future directions are proposed.
