  We examine the structure of two possible candidates of isometry groups for
the spectral triples on $AF$-algebras introduced by Christensen and Ivan. In
particular, we completely determine the isometry group introduced by Park, and
observe that these groups coincide in the case of the Cantor set. We also show
that the construction of spectral triples on crossed products given by Hawkins,
Skalski, White and Zacharias, is suitable for the purpose of lifting
isometries.
