In this paper we establish the action of the Grothendieck-Teichmüller group $\widehat{GT}$ on the prime order torsion elements of the profinite fundamental group ${\pi }_1^{geom}\left({ℳ}_{0,\left[n\right]}\right)$. As an intermediate result, we prove that the conjugacy classes of prime order torsion of ${\widehat{\pi }}_1\left({ℳ}_{0,\left[n\right]}\right)$ are exactly the discrete prime order ones of the ${\pi }_1\left({ℳ}_{0,\left[n\right]}\right)$.