  Associated to every finite group, Kitaev has defined the quantum double model
for every orientable surface without boundary. In this paper, we define
boundaries for this model and characterize condensations; that is, we find all
quasi-particle excitations (anyons) which disappear when they move to the
boundary. We then consider two phases of the quantum double model corresponding
to two groups with a domain wall between them, and study the tunneling of
anyons from one phase to the other. Using this framework we discuss the
necessary and sufficient conditions when two different groups give the same
anyon types. As an application we show that in the quantum double model for S_3
(the permutation group over three letters) there is a chargeon and a fluxion
which are not distinguishable. This group is indeed a special case of groups of
the form of the semidirect product of the additive and multiplicative groups of
a finite field, for all of which we prove a similar symmetry.
