  We study those automatic sequences which are produced by an automaton whose
underlying graph is the Cayley graph of a finite group. For $2$-automatic
sequences, we find a characterization in terms of what we call homogeneity, and
among homogeneous sequences, we single out those enjoying what we call
self-similarity. It turns out that self-similar $2$-automatic sequences (viewed
up to a permutation of their alphabet) are in bijection with many interesting
objects, for example dessins d'enfants (covers of the Riemann sphere with three
points removed).
  For any $p$ we show that, in the case of an automatic sequence produced "by a
Cayley graph", the group and indeed the automaton can be recovered canonically
from the sequence.
  Further, we show that a rational fraction may be associated to any automatic
sequence. To compute this fraction explicitly, knowledge of a certain graph is
required. We prove that for the sequences studied in the first part, the graph
is simply the Cayley graph that we start from, and so calculations are
possible.
  We give applications to the study of the frequencies of letters.
