  Let $A$ denote the alternation limit algebra, studied by Hopenwasser and
Power, and by Poon, which is the closed direct limit of upper triangular matrix
algebras determined by refinement embeddings of multiplicity $r_k$ and standard
embeddings of multiplicity $s_k$. It is shown that the quotient of the
isometric automorphism group by the approximately inner automorphisms is the
abelian group $ \ZZ ^d$ where $d$ is the number of primes that are divisors of
infinitely many terms of each of the sequences $(r_k)$ and $(s_k)$. This group
is also the group of automorphisms of the fundamental relation of $A$.
