  It is known that if an Abelian group operation is used in an Ar{\i}kan-style
construction, we have multilevel polarization where synthetic channels can
approach intermediate channels that are neither almost perfect nor almost
useless. An open problem in polarization theory is to determine the
polarization levels of a given channel. In this paper, we discuss the
polarization levels of a family of channels that we call automorphic-symmetric
channels. We show that the polarization levels of an automorphic-symmetric
channel are determined by characteristic subgroups. In particular, if the group
that is used does not contain any non-trivial characteristic subgroup, we only
have two-level polarization to almost perfect and almost useless channels.
