Abstract
In this article we completely characterise constant length substitution shifts which have a proper almost automorphic factor, or which have a bijective substitution factor such that the factor map is injective on at least one point. Our approach is algebraic: we characterise these dynamical properties in terms of a finite semigroup defined by the substitution. We characterise the existence of almost automorphic factors in terms of Green's R -relation and the existence of bijective factors in terms of Green's L -relation. Our results are constructive.