An avoshift is a subshift where for each set C from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on C to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets C) by, for example, cellwise quasigroup shifts, totally extremally permutive (TEP) subshifts, and subshifts of finite type (SFTs) with a safe symbol. In this paper, we concentrate on avoshifts on polycyclic groups, when the sets C are what we call 'inductive intervals'. We show that then, avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.