We consider the question of when, given a subset $A$ of $M$, the setwise stabilizer of the group of automorphisms induces a closed subgroup on $\mathrm{Sym}(A)$. We define s-homogeneity to be the analogue of homogeneity relative to strong embeddings and show that any subset of a countable, s-homogeneous, $\omega$-stable structure induces a closed subgroup and contrast this with a number of negative results. We also show that for $\omega$-stable structures s-homogeneity is preserved under naming countably many constants, but under slightly weaker conditions it can be lost by naming a single point.