We construct the fundamental solutions
Γ
\Gamma
and
γ
\gamma
for the non-divergence form operators
∑
i
,
j
a
i
,
j
(
x
,
t
)
X
i
X
j
−
∂
t
\,{\textstyle \sum _{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial _t\,
and
∑
i
,
j
a
i
,
j
(
x
)
X
i
X
j
{\,\textstyle \sum _{i,\,j}}\,a_{i,\,j}(x)\,X_iX_j
, where the
X
i
X_i
’s are Hörmander vector fields generating a stratified group
G
\mathbb {G}
and
(
a
i
,
j
)
i
,
j
(a_{i,j})_{i,j}
is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of
Γ
\Gamma
and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of
Γ
\Gamma
allow us to construct
γ
\gamma
using both
t
t
-saturation and approximation arguments.