  The boundary of a fractionalized topological phase can be gapped by
condensing a proper set of bosonic quasiparticles. Interestingly, in the
presence of a global symmetry, such a boundary can have different symmetry
transformation properties. Here we present an explicit example of this kind, in
the double semion state with time reversal symmetry. We find two distinct cases
where the semionic excitations on the boundary can transform either as time
reversal singlets or as time reversal (Kramers) doublets, depending on the
coherent phase factor of the Bose condensate. The existence of these two
possibilities are demonstrated using both field theory argument and exactly
solvable lattice models. Furthermore, we study the domain walls between these
two types of gapped boundaries and find that the application of time reversal
symmetry tunnels a semion between them.
