  We exhibit an infinite family of {\it triplets} of mutually unbiased bases
(MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard
matrices, $F(a,b)$. However, in the main result of the paper we also prove that
for any values of the parameters $(a,b)$, the standard basis and $F(a,b)$ {\it
cannot be extended to a MUB-quartet}. The main novelty lies in the {\it method}
of proof which may successfully be applied in the future to prove that the
maximal number of MUBs in dimension 6 is three.
