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Just what it says on the tin:

For a set, X, is there a word to describe the union of sets of permutations of each member of the powerset of X?

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    Glad I could help. If my answer is satisfactory, you can "accept" it by clicking the checkmark just underneath the number and the up and down arrows.2012-06-14

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Your phrasing is a little unclear. If $X$ is our set and $\mathcal{P}(X)$ is its power set, I think you either mean $\operatorname{Aut}(\mathcal{P}(X))=\{\text{permutations of the set }\mathcal{P}(X)\}$ or $\bigcup_{S\in \mathcal{P}(X)}\operatorname{Aut}(S)=\bigcup_{S\in\mathcal{P}(X)}\{\text{permutations of the set }S\}$ In the first case, I would just call it "permutations of the set of subsets of $X$", while in the second case, this set appears to be known as the set of partial permutations of $X$.