  An extension of subgroups $H\leqslant K\leqslant F_A$ of the free group of
rank $|A|=r\geqslant 2$ is called onto when, for every ambient free basis $A'$,
the Stallings graph $\Gamma_{A'}(K)$ is a quotient of $\Gamma_{A'}(H)$.
Algebraic extensions are onto and the converse implication was conjectured by
Miasnikov-Ventura-Weil, and resolved in the negative, first by
Parzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank.
In this note we study properties of this new type of extension among free
groups (as well as the fully onto variant), and investigate their corresponding
closure operators. Interestingly, the natural attempt for a dual notion -- into
extensions -- becomes trivial, making a Takahasi type theorem not possible in
this setting.
