  $H$-type foliations $(\mathbb{M},\mathcal{H},g_{\mathcal{H}})$ are studied in
the framework of sub-Riemannian geometry with bracket generating distribution
defined as the bundle transversal to the fibers. Equipping $\mathbb{M}$ with
the Bott connection we consider the scalar horizontal curvature
$\kappa_{\mathcal{H}}$ as well as a new local invariant $\tau_{\mathcal{V}}$
induced from the vertical distribution. We extend recent results on the
small-time asymptotics of the sub-Riemannanian heat kernel on
quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second
heat invariant in sub-Riemannian geometry as a linear combination of
$\kappa_{\mathcal{H}}$ and $\tau_{\mathcal{V}}$. The use of an analog to normal
coordinates in Riemannian geometry that are well-adapted to the geometric
structure of $H$-type foliations allows us to consider the pull-back of
Kor\'{a}nyi balls to $\mathbb{M}$. We explicitly obtain the first three terms
in the asymptotic expansion of their Popp volume for small radii. Finally, we
address the question of when $\mathbb{M}$ is locally isometric as a
sub-Riemannian manifold to its $H$-type tangent group.
