Onto extensions of free groups

An extension of subgroups $H\leqslant K\leqslant F_A$ of the free group ofrank $|A|=r\geqslant 2$ 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 byMiasnikov-Ventura-Weil, and resolved in the negative, first byParzanchevski-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 freegroups (as well as the fully onto variant), and investigate their correspondingclosure operators. Interestingly, the natural attempt for a dual notion -- intoextensions -- becomes trivial, making a Takahasi type theorem not possible inthis setting.