Detection of non-orientability by solvable subgroups

Determine whether, for every finite group G and every real representation V of G, non-orientability of the representation sphere S^V implies that the parity function π_V(H) = dim(V^H) mod 2 is nonconstant on the solvable subgroups of G.

Background

The paper proves that if the parity function of a real representation is nonconstant on the solvable subgroups of a finite group, then the representation sphere is non-orientable. Equivalently, a solvable pair of subgroups K_1 normal in K_2 with index two detects a parity change and hence non-orientability.

The unresolved direction asks whether every non-orientable representation sphere must already be detected in this way on solvable subgroups. Computational tests found no counterexample among groups of order at most 2000 and among the finite simple groups with available TomLib data, but the authors explain that the difficulty is concentrated in nonsolvable, especially perfect, groups.