  We study the automorphism group of an infinite minimal shift $(X,\sigma)$
such that the complexity difference function, $p(n+1)-p(n)$, is bounded. We
give some new bounds on $\mbox{Aut}(X,\sigma)/\langle \sigma \rangle$ and also
study the one-sided case. For a class of Toeplitz shifts, including the class
of shifts defined by constant length primitive substitutions with a coincidence
and with height one, we show that the two-sided automorphism group is a cyclic
group. We next focus on shifts generated by primitive constant length
substitutions. For these shifts, we give an algorithm that computes their
two-sided automorphism group, As a corollary we describe how to compute the set
of conjugacies between two such shifts.
