  The quantum symmetry group of the inductive limit of C*-algebras equipped
with orthogonal filtrations is shown to be the projective limit of the quantum
symmetry groups of the C*-algebras appearing in the sequence. Some explicit
examples of such projective limits are studied, including the case of quantum
symmetry groups of the duals of finite symmetric groups, which do not fit
directly into the framework of the main theorem and require further specific
study. The investigations reveal a deep connection between quantum symmetry
groups of discrete group duals and the doubling construction for Hopf algebras.
