  We study strongly harmonic functions in Carnot-Carath\'eodory groups defined
via the mean value property with respect to the Lebesgue measure. For such
functions we show their Sobolev regularity and smoothness. Moreover, we prove
that strongly harmonic functions satisfy the sub-Laplace equation for the
appropriate gauge norm and that the inclusion is sharp. We observe that
spherical harmonic polynomials in $\mathbb{H}_1$ are both strongly harmonic and
satisfy the sub-Laplace equation. Our presentation is illustrated by examples.
