There is a proved extension beyond \(C=1\) when the prime-to-\(p\) stabilizer acts trivially on the source extension and its honest operators on the entire rank-\(N\) source carrier are \(\chi(c)I_N\) in the actual inertia image.
Under the same-source determinant central-character condition, the required scalar phase can be implemented by actual central matrices after a finite cover \(\widetilde C\twoheadrightarrow C\). The kernel acts trivially on the whole recovered candidate, so the represented action descends to \(C\).
Reading the diagram
A cover may be needed even when the represented operator is a scalar. A direct matrix homomorphism from \(C\) need not exist.
Scope
The theorem requires a genuine inertia phase on the whole carrier, the fixed reciprocity convention, and compatible completion. It does not apply merely because an arbitrary abstract phase on a multiplicity subspace is scalar. It does not change a fixed canonical beta extension to force a desired action.