Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth

Let $\mathcal{L}$ be a sub-Laplacian on a connected Lie group $G$ of polynomial growth. It is well known that, if $F:\mathbb{R}\to \mathbb{C}$ is in the Schwartz class $\mathcal{S}(\mathbb{R})$, then the convolution kernel ${\mathcal{K}}_{F(\mathcal{L})}$ of the operator $F(\mathcal{L})$ is in the Schwartz class $\mathcal{S}(G)$. Here we prove a sort of converse implication for a class of groups $G$ including all solvable noncompact groups of polynomial growth. We also discuss the problem whether integrability of ${\mathcal{K}}_{F(\mathcal{L})}$ implies continuity of $F$.