Non-standard sequence subgroups in finite fields

In previous papers, see, for example, [O.J. Brison, J.E. Nogueira, Second order linear sequence subgroups in finite fields, Finite Fields Appl. 14 (2008) 277–290], the authors studied f-sequence subgroups when f(t) was a polynomial of degree 2. Here we study non-standard groups when f(t) has degree k. The non-standard configurations found when k=2 extend to the general case, while new configurations when k>2 are described.