Let $F$ be a set and $.$ be a binary operation on $F$ and $.^{-1}:F\to F$ be a so-called inverse operation on $F$ such that $(F,.)$ is semigroup and for each $x,y\in F$,
$$(x^{-1})^{-1}=x,~~~~~ (xy)^{-1}=y^{-1}x^{-1}$$
Then $(F,.,^{-1})$ is called an Ef Stucture.
An ef structure is called derived if it is isomorphic to some $(\mathcal F,.,^{-1})$ where $\mathcal F\subseteq \mathcal P(G)$ for a group $G$.
Which ef structures are derived? Is there an algebraic (structural) characterization?