We investigate the structure of the automorphism groups of Kimura Hadamard matrices (KHMs) constructed from dihedral groups. We identify several different types of automorphisms and show that the automorphism group of a KHM always has a subgroup isomorphic to \(D_{2k}\times Q_8\) , or \(C_2\times D_{2k}\times Q_8\) if it is y-invariant. We exhibit additional automorphisms arising from the holomorph of the dihedral group under suitable structural conditions. A comparison with known examples, including those of Kimura, Niwasaki, and matrices arising from the Shinoda–Yamada construction, reveals counterexamples to a conjecture of Ó Cathín and suggests that no further automorphisms occur beyond those predicted by our framework.
