  We calculate the exact amenability constant of the centre of $\ell^1(G)$ when
$G$ is one of the following classes of finite group: dihedral; extraspecial; or
Frobenius with abelian complement and kernel. This is done using a formula
which 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 (J. Funct. Anal. 2009).
