  We compute the sets of degrees of maps between principal $SU(2)$-bundles over
$S^5$, i.e. between any of the manifolds $SU(2)\times S^5$ and $SU(3)$. We show
that the Steenrod squares provide the only obstruction to the existence of a
mapping degree between these manifolds, and construct explicit maps realizing
each integer that occurs as a mapping degree.
  Added Erratum. After this manuscript was accepted for publication by
Transformation Groups, Xueqi Wang pointed out a mistake in our paper. At the
end of this arXiv version we add an erratum, where we correct the statement and
the proof of Theorem 1.1.
