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We construct the fundamental solutions $\Gamma$ and $\gamma$ for the non-divergence form operators ${\textstyle\sum_{i,j}} a_{i,j}(x,t)X_iX_j-\partial_t$ and ${\textstyle\sum_{i,j}}a_{i,j}(x)X_iX_j$, where the $X_i$'s are Hörmander vector fields generating a stratified group $\mathbb{G}$ and $(a_{i,j})_{i,j}$ is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of $\Gamma$ and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of $\Gamma$ allow us to construct $\gamma$ using both $t$-saturation and approximation arguments. [ABSTRACT FROM AUTHOR]