  Recent constructions have shown that interesting behaviours can be observed
in the finiteness properties of K\"ahler groups and their subgroups. In this
work, we push this further and exhibit, for each integer $k$, new hyperbolic
groups admiting surjective homomorphisms to $\mathbb{Z}$ and to
$\mathbb{Z}^{2}$, whose kernel is of type $\mathscr{F}_{k}$ but not of type
$\mathscr{F}_{k+1}$. By a fibre product construction, we also find examples of
nonnormal subgroups of K\"ahler groups with exotic finiteness properties.
