Denote by $\Bbb A$ the class of all absolutely continuous contractions whose associated Sz. Nagy-Foias functional calculus is isometric. Starting from the factthat if $u$ is a non-constant inner function and if$T\in {\Bbb A}$, then so does $u(T)$, we study howinner functions operate on the classes${\Bbb A}_{n,m}$, subclasses of the class ${\Bbb A}$.For this purpose, we use standard dual algebra techniquesand a decomposition of the algebra $H^\infty$ into aweak*-topological direct sum of copies of itself.We also discuss mapping theorems for the support of the spectral measures associated with the unitary parts of theminimal isometric extension and the minimal co-isometricextensionofan absolutely continuous contraction $T$.