Non-permutation invariant Borel quantifiers

Every permutation invariant Borel subset of the space of countable structures is definable in $\La_{\omega_1\omega}$ by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed subgroup $G$ of the symmetric group $S_{\infty}$, there is a closed binary quantifier $Q$ such that the $G$-invariant subsets of the space of countable structures are exactly the $\La_{\omega_1\omega}(Q)$-definable sets.