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Let $${\mathbb{G}}$$ be a Carnot group of step r and m generators and homogeneous dimension Q. Let $${\mathbb{F}_{m,r}}$$ denote the free Lie group of step r and m generators. Let also $${\pi:\mathbb{F}_{m,r}\to\mathbb{G}}$$ be a lifting map. We show that any horizontally convex function u on $${\mathbb{G}}$$ lifts to a horizontally convex function $${u\circ \pi}$$ on $${\mathbb{F}_{m,r}}$$ (with respect to a suitable horizontal frame on $${\mathbb{F}_{m,r}}$$ ). One of the main aims of the paper is to exhibit an example of a sub-Laplacian $${\mathcal{L}=\sum_{j=1}^m X_j^2}$$ on a Carnot group of step two such that the relevant $${\mathcal{L}}$$ -gauge function d (i.e., d 2-Q is the fundamental solution for $${\mathcal{L}}$$ ) is not h-convex with respect to the horizontal frame {X 1, . . . , X m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).