  We perform a study of time reversal symmetry of abelian anyons $\mathcal{A}$
in 2+1 dimensions, in the spin structure independent cases. We will find the
importance of the group $\mathcal{C}$ of time-reversal-symmetric anyons modulo
anyons composed from an anyon and its time reversal. Possible choices of local
Kramers degeneracy are given by quadratic refinements of the braiding phases of
$\mathcal{C}$, and the anomaly is then given by the Arf invariant of the chosen
quadratic refinement. We also give a concrete study of the cases when
$|\mathcal{A}|$ is odd or $\mathcal{A}=(\mathbb{Z}_2)^N$.
