We calculate the exact amenability constant of the centre of
${{\ell }^{1}}\left( G \right)$ when
$G$ is a finite group and is either dihedral, extraspecial, or Frobenius with abelian complement and kernel. This is done using a formula that applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk.