Higher incoherence of the automorphism groups of a free group

Let $F_n$ be the free group on $n\ge 2$ generators. We show that for all $1\le m\le 2n-3$ (respectively for all $1\le m\le 2n-4$) there exists a subgroup of $\mathrm{Aut}(F_n)$ (respectively $\mathrm{Out}(F_n)$) which has finiteness of type $F_m$ but not of type $F{P}_{m+1}(\mathbb{Q})$, hence it is not $m$-coherent. In both cases, the new result is the upper bound $m=2n-3$ (respectively $m=2n-4$), as it cannot be obtained by embedding direct products of free noncyclic groups, and certifies higher incoherence up to the virtual cohomological dimension and is therefore sharp. As a tool of the proof, we discuss the existence and nature of multiple inequivalent extensions of a suitable finite-index subgroup $K_4$ of $\mathrm{Aut}(F_2)$ (isomorphic to the quotient of the pure braid group on four strands by its center): the fiber of four of these extensions arise from the strand-forgetting maps on the braid groups, while a fifth is related with the Cardano-Ferrari epimorphism.