  In this paper we study nonlocal equations driven by the fractional powers of
hypoelliptic operators in the form $$\mathscr K u = \mathscr A u - \partial_t u
\overset{def}{=} \operatorname{tr}(Q \nabla^2 u) + <BX,\nabla u> - \partial_t
u,$$ introduced by H\"ormander in his 1967 hypoellipticity paper. We show that
the nonlocal operators $(-\mathscr K)^s$ and $(-\mathscr A)^s$ can be realized
as the Dirichlet-to-Neumann map of doubly-degenerate extension problems. We
solve such problems in $L^\infty$, and in $L^p$ for $1\leq p<\infty$ when
$\operatorname{tr}(B)\geq 0$. In forthcoming works we use such calculus to
establish some new Sobolev and isoperimetric inequalities.
