Images of word maps in finite simple groups

In response to questions by Kassabov, Nikolov and Shalev, we show that a given subset $A$ of a finite simple group $G$ is the image of some word map $w : G\times G\to G $ifandonlyif(i)$A$containstheidentityand(ii)$A$ is invariant under $\text{Aut}(G)$.