  In this article we continue the study of automorphism groups of constant
length substitution shifts and also their topological factors. We show that up
to conjugacy, all roots of the identity map are letter exchanging maps, and all
other nontrivial automorphisms arise from {\em twisted} compressions of another
constant length substitution. We characterise the group of roots of the
identity in both the measurable and topological setting. Finally, we show that
any topological factor of a constant length substitution shift is topologically
conjugate to a constant length substitution shift via a letter-to-letter code.
