  Let $F_n$ be the free group on $n \geq 2$ generators. We show that for all $1
\leq m \leq 2n-3$ (respectively for all $1 \leq m \leq 2n-4$) there exists a
subgroup of $\operatorname{Aut}(F_n)$ (respectively $\operatorname{Out}(F_n)$)
which has finiteness of type $F_{m}$ but not of type $FP_{m+1}(\mathbb{Q})$,
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 suitable finite-index subgroups of $\operatorname{Aut}(F_2)$.
