aperiodic order with tetrahedral symmetry. Specifically, we consider two distinct crystallographic representations of the icosahedral group sharing a common maximal subgroup ${\mathcal{G}}$, and define rotations in $SO(6)$ that keep the ${\mathcal{G}}$-symmetry preserved. Such rotations induce, in projection, continuous ${\mathcal{G}}$-symmetry preserving transformations of the corresponding model sets or point arrays. We show that these rotations are parameterised by angles belonging to a $k$-dimensional torus $\mathbb{T}^k$, which are candidates to be the order parameters of the Landau theory. Moreover, we classify, based on the results of our previous paper [@zappa1], the possible boundary conditions of the transitions, which, in this context, correspond to specific angles in $\mathbb{T}^k$.

Structural transitions of quasicrystals and point arrays have been analysed previously with different approaches, for examples with the phason strain approach [@cheng], which was later proved to be equivalent to the Schur rotation method [@torres]. Moreover, Indelicato *et al.* [@paolo; @giuliana] define higher dimensional generalisations of the Bain strain between crystal lattices which induce, in projection, structural transformations of quasicrystals. In the latter reference it is proved that the Schur rotation method is related to the Bain strain in the case of transitions between cubic to icosahedral order [@paolo], and some specific examples are given 