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On $\mathbb{R}^n,$ a classical result due to Bourgain establishes therestricted weak $(\frac{n}{n-1},1)$ inequality for the full maximal function$M_F^{d\sigma}$ associated to the spherical averages. In this work we presentan extension to Bourgain's result on graded Lie groups for a family of fullmaximal operators. We formulate this extension using the group Fouriertransform of the measures under consideration and the symbols of (positiveRockland operators which are) positive left-invariant hypoelliptic partialdifferential operators on graded Lie groups.

Estimates for the full maximal function on graded Lie groups