  We consider continuous, translation-commuting transformations of compact,
translation-invariant families of mappingsfrom finitely generated groups into
finite alphabets. It is well-known that such transformations and spaces can be
described "locally" via families of patterns and finitary functions; such
descriptions can be re-used on groups larger than the original, usually
defining non-isomorphic structures. We show how some of the properties of the
"induced" entities can be deduced from those of the original ones, and vice
versa; then, we show how to "simulate" the smaller structure into the larger
one, and obtain a characterization in terms of group actions for the dynamical
systems admitting of presentations via structures as such. Special attention is
given to the class of sofic shifts.
