Given a group $G$ and a subset of the group $S$, the normalizer $N_G(S)$ is defined as the set $$ N_G(S) = \{ g\in G \mid \forall s\in S, \ gsg^{-1}\in S \} $$
Is a weaker version of that (which is certainly not a group), such that $$ W_G(S) = \{ g\in G \mid \exists s\in S, \ gsg^{-1}\in S \} $$ well-known and studied?