  We propose a new method to construct degenerate
$3$-$(\alpha,\delta)$-Sasakian manifolds as fiber products of Boothby-Wang
bundles over hyperk\"ahler manifolds. Subsequently, we study homogeneous
degenerate $3$-$(\alpha, \delta)$-Sasakian manifolds and prove that no
non-trivial compact examples exist as well as that there is exactly one family
of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.
