  Let $G = N \rtimes A$, where $N$ is a stratified Lie group and $A= \mathbb
R_+$ acts on $N$ via automorphic dilations. We prove that the group $G$ has the
Calder\'on-Zygmund property, in the sense of Hebisch and Steger, with respect
to a family of flow measures and metrics. This generalizes in various
directions previous works by Hebisch and Steger and Martini, Ottazzi and
Vallarino, and provides a new approach in the development of Calder\'on-Zygmund
theory in Lie groups of exponential growth. We also prove a weak type $(1,1)$
estimate for the Hardy-Littlewood maximal operator naturally arising in this
setting.
