  We study the equivariant flag $f$-vector and equivariant flag $h$-vector of a
balanced relative simplicial complex with respect to a group action. When the
complex satisfies Serre's condition $(S_{\ell}),$ we show that the equivariant
flag $h$-vector, the equivariant $h$-vector, and the equivariant $f$-vector
satisfy several inequalities.
  We apply these results to the study of $P$-partitions of double posets, and
weak colorings of mixed graphs.
