Given the normalizer N(G) of a group $G < S_n$, is $G$ uniquely defined? In either case is there a procedure to find "a" $G < S_n$ with given normalizer $N(G)$?
Does such $G$ always EXIST?
The normalizer is defined as $N_{S_n}(G) = \{\pi \in S_n | \pi G \pi^{-1} = G\}$
Added followup questions:
Is there a property that guarantees a given group $H<S_n$ is normalizer for some other group? that is $$property(H) \quad \Rightarrow \quad \exists G < S_n \quad \text{s.t.} H = N(G)$$