  Jet spaces on $\mathbb R^n$ have been shown to have a canonical structure of
stratified Lie groups (also known as Carnot groups). We construct jet spaces
over stratified Lie groups adapted to horizontal differentiation and show that
these jet spaces are themselves stratified Lie groups. Furthermore, we show
that these jet spaces support a prolongation theory for contact maps, and in
particular, a B\"acklund type theorem holds. A byproduct of these results is an
embedding theorem that shows that every stratified Lie group of step $s+1$ can
be embedded in a jet space over a stratified Lie group of step $s$.
