  We consider the category of finite dimensional representations of the quantum
double of a finite group as a modular tensor category. We study
auto-equivalences of this category whose induced permutations on the set of
simple objects (particles) are of the special form of PJ, where J sends every
particle to its charge conjugation and P is a transposition of a
chargeon-fluxion pair. We prove that if the underlying group is the semidirect
product of the additive and multiplicative groups of a finite field, then such
an auto-equivalence exists. In particular, we show that for S_3 (the
permutation group over three letters) there is a chargeon and a fluxion which
are not distinguishable. Conversely, by considering such permutations as
modular invariants, we show that a transposition of a chargeon-fluxion pair
forms a modular invariant if and only if the corresponding group is isomorphic
to the semidirect product of the additive and multiplicative groups of a finite
near-field.
