  In combinatorial topology we aim to triangulate manifolds such that their
topological properties are reflected in the combinatorial structure of their
description. Here, we give a combinatorial criterion on when exactly
triangulations of 3-manifolds with transitive cyclic symmetry can be
generalised to an infinite family of such triangulations with similarly strong
combinatorial properties. In particular, we construct triangulations of Seifert
fibred spaces with transitive cyclic symmetry where the symmetry preserves the
fibres and acts non-trivially on the homology of the spaces. The triangulations
include the Brieskorn homology spheres $\Sigma (p,q,r)$, the lens spaces
$\operatorname{L} (q,1)$ and, as a limit case, $(\mathbf{S}^2 \times
\mathbf{S}^1)^{\# (p-1)(q-1)}$.
