In this paper the authors consider a class of second order ultraparabolic differential equations of the form \[ L_Au:=\sum_{j,k=1}^mX_j(a_{jk}X_ku)+X_0u-\partial_tu=0, \] where \(A=(a_{jk})\) is a bounded symmetric, uniformly positive matrix, the operator \[ \sum_{j=1}^mX_j^2+X_0-\partial_t \] is supposed to be hypoelliptic, and where the (smooth) vectors fields \(X_1,\ldots,X_m\) and \(Y:=X_0-\partial_t\) are supposed to be invariant with respect to a suitable homogeneous Lie group \(G\). In the assumption that the entries \(a_{jk}\) be measurable, the authors adapt Moser’s iterative method to the non-Euclidean geometry of \(G\) to prove an \(L^\infty_{\text{loc}}\) bound of the solution \(u\) in terms of the \(L^p_{\text{loc}}\) norm of \(u.\) They next prove a pointwise upper bound for the fundamental solution of \(L_A\) given in terms of the value-function of an optimal control problem related to the vector fields \(X_1,\ldots,X_m,Y\), and finally, on supposing that the entries \(a_{jk}\) be smooth, they show the existence of the fundamental solution of \(L_A\), giving a precise description of its properties.