Which homotopy algebras come from transfer?

We characterize A∞A_\infty-structures that are equivalent to a given transferred structure over a chain homotopy equivalence or a quasi-isomorphism, answering a question posed by D. Sullivan. Along the way, we present an obstruction theory for weak \text{A∞A_\infty-morphisms} over an arbitrary commutative ring. We then generalize our results to P∞\mathcal {P}_\infty-structures over a field of characteristic zero, for any quadratic Koszul operad P\mathcal {P}.