  It was shown in [A. Azimifard, E. Samei, N. Spronk, JFA 2009; arxiv
0805.3685] that the ZL-amenability constant of a finite group is always at
least 1, with equality if and only if the group is abelian. It was also shown
in the same paper that for any finite non-abelian group this invariant is at
least 301/300, but the proof relies crucially on a deep result of D. A. Rider
on norms of central idempotents in group algebras.
  Here we show that if G is finite and non-abelian then its ZL-amenability
constant is at least 7/4, which is known to be best possible. We avoid use of
Rider's result, by analyzing the cases where G is just non-abelian, using
calculations from [M. Alaghmandan, Y. Choi, E. Samei, CMB 2014; arxiv
1302.1929], and establishing a new estimate for groups with trivial centre.
