Reduction of nirrorno-ness along odd-index normal subgroups

Establish whether an analogue of the strong nirrorno reduction theorem holds for nirrorno-ness: namely, determine whether a finite group G is nirrorno whenever it has a normal nirrorno subgroup N of odd index [G:N].

Background

The paper proves that if N is a normal subgroup of a finite group G with odd index and N is strongly nirrorno, then G is strongly nirrorno. This yields, in particular, strong nirrorno-ness for groups with a normal Sylow 2-subgroup.

The authors explicitly note that their reduction argument relies on linear independence of parity functions, a condition available for strong nirrorno-ness but not known to follow from ordinary nirrorno-ness. The open issue is therefore whether the same odd-index inheritance principle can be proved without the stronger linear-independence hypothesis.