  We describe a set of conformally covariant boundary operators associated to
the sixth-order GJMS operator on a conformally invariant class of manifolds
which includes compactifications of Poincar\'e--Einstein manifolds. This yields
a conformally covariant energy functional for the sixth-order GJMS operator on
such manifolds. Our boundary operators also provide a new realization of the
fractional GJMS operators of order one, three, and five as generalized
Dirichlet-to-Neumann operators. This allows us to prove some sharp Sobolev
trace inequalities involving the interior $W^{3,2}$-seminorm, including an
analogue of the Lebedev--Milin inequality on six-dimensional manifolds.
