Second order linear sequence subgroups in finite fields

In the previous paper [O.J. Brison, J.E. Nogueira, Linear recurring sequence subgroups in finite fields, Finite Fields Appl. 9 (2003) 413–422] the authors investigated when, and how, the elements of a multiplicative subgroup of a finite field can be written, without repetition, as a cyclically-closed second order linear recurring sequence. Here, the earlier results are extended; in particular a general method is suggested for attempting to prove a given subgroup to be “standard,” in the terminology of the previous paper, while a lifting theorem for “nonstandard” subgroups is proved.