A Semi-Abelian Group of Order 768 That Is Not an M-Group

M. Kida conjectured that every finite semi-abelian group is monomial (J. Group Theory, 2025; Kourovka Notebook, Problem 21.68). We show that the conjecture is false. If a finite group \(W\) acts on a finite abelian group \(N\), a linear character of \(N\) has stabiliser \(T\) in \(W\), and \(T\) has an irreducible character that is not monomial, then \(N\rtimes W\) is not an M-group. We apply this to the augmentation submodule of the permutation module \(\mathbb{F}_2[W/T]\), where \(W=E\rtimes A_4\) is the index-two subgroup of \(C_2\wr A_4\) and \(T\cong\mathrm{SL}(2,3)\) is the binary tetrahedral group acting on the eight quaternion units. The result is a semi-abelian group of order \(768=2^8\cdot 3\) with a non-monomial irreducible character of degree \(8\). The proof uses only Clifford theory. The example was also checked by exact computation, including every subgroup of index \(8\). This is an unrefereed note; minimality of the order is not claimed.