  We study the family of vertical projections whose fibers are right cosets of
horizontal planes in the Heisenberg group, $\mathbb{H}^n$. We prove lower
bounds for Hausdorff dimension distortion of sets under these mappings, with
respect to the Euclidean metric and also the natural quotient metric which we
show behaves like the Euclidean metric in this context. Our bounds are sharp in
a large part of the dimension range, and we give conjectural sharp lower bounds
for the remaining range. Our approach also lets us improve the known almost
sure lower bound for the standard family of vertical projections in
$\mathbb{H}^n$ for $n \geq 2$.
