Conjugacy Action on Subsets is Group Action

Theorem
Let $\powerset G$ be the set of all subgroups of $G$.

For any $S \in \powerset G$ and for any $g \in G$, the conjugacy action:
 * $g * S := g \circ S \circ g^{-1}$

is a group action.

Proof
By, $e \in G$, thus:

Thus is seen to be fulfilled.

Then:

Thus is shown to be fulfilled.