  We prove that the symmetrizer of a permutation group preserves stability of a
polynomial if and only if the group is orbit homogeneous. A consequence is that
the hypothesis of permutation invariance in the Grace-Walsh-Szeg\H{o}
Coincidence Theorem cannot be relaxed. In the process we obtain a new
characterization of the \emph{Grace-like polynomials} introduced by D. Ruelle,
and prove that the class of such polynomials can be endowed with a natural
multiplication.
