  Mutually unbiased bases (MUB) are an elusive discrete structure in Hilbert
spaces. Many (complete sets of) MUB are group covariant, but little is known
whether they can be sharply covariant in the sense that the generating groups
can have order equal to the total number of basis states, that is, $d(d+1)$ for
MUB in dimension $d$. Sharply covariant MUB, if they exist, would be most
appealing from both theoretical and practical point of view. Since stabilizer
MUB subsume almost all MUB that have ever been constructed, it is of
fundamental interest to single out those candidates that are sharply covariant.
We show that, quite surprisingly, only two stabilizer MUB are sharply
covariant, and the conclusion still holds even if antiunitary transformations
are taken into account. Our study provides valuable insight on the symmetry of
stabilizer MUB, which may have implications for a number of research topics in
quantum information and foundations. In addition, it exposes a sharp contrast
between MUB and another elusive discrete structure known as symmetric
informationally complete measurements (SICs), all known examples of which are
sharply covariant.
