Here is a theorem of Hjorth-Kechris-Louveau which might be of some interest.
Theorem: Let $E^X_G$ be the orbit equivalence relation induced by a Borel action of a closed subgroup $G$ of $Sym(\mathbb{N})$ on some standard Borel space $X$. Then the following are equivalent:
i. $E^X_G$ is essentially countable.
ii. For some Polish topology $\tau$ on $X$ giving its Borel structure, $E^X_G$ is $\mathbf{\Sigma^0_2}$ in $(X,\tau)^2$.
ii. For some Polish topology $\tau$ on $X$ giving its Borel structure, $E^X_G$ is $\mathbf{\Sigma^0_3}$ in $(X,\tau)^2$.
(This is Theorem 1.12 in Jackson, Kechris and Louveau's paper "Countable Borel equivalence relations". The following two papers are given as references: Borel equivalence relations induced by actions of the symmetric group, Borel equivalence relations and classifications of countable models)