  We introduce the anonymity group, the neutrality group and the symmetry group
of a social preference function. Inspired by a problem posed by Kelly in 1991
and remained unsolved, we investigate the problem of recognizing which
permutation groups may arise as anonymity, neutrality and symmetry group of a
social preference function.
  A complete description is found for the neutrality groups and a sufficient
condition, which largely encompasses the problem, is found for the anonymity
groups.
  Using the concept of orbit extension of a group $U$, we formulate manageable
necessary conditions for being $U$ an anonymity or a symmetry group. Our
research deeply interacts with problems of representability by Boolean
functions shedding light on them.
