  A quasisymmetric function is assigned to every double poset (that is, every
finite set endowed with two partial orders) and any weight function on its
ground set. This generalizes well-known objects such as monomial and
fundamental quasisymmetric functions, (skew) Schur functions, dual immaculate
functions, and quasisymmetric $\left(P, \omega\right)$-partition enumerators.
We prove a formula for the antipode of this function that holds under certain
conditions (which are satisfied when the second order of the double poset is
total, but also in some other cases); this restates (in a way that to us seems
more natural) a result by Malvenuto and Reutenauer, but our proof is new and
self-contained. We generalize it further to an even more comprehensive setting,
where a group acts on the double poset by automorphisms.
