Onto extensions of free groups

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