Let G be a group. Let $G^p$ be the completion of G with respect to the mod p lower central series of G.i.e. $G^p=\varprojlim_{q} G/\gamma_qG$, where $\gamma_qG$ is generated by all $\{[x_1,\cdots,x_s]^{p^t}:sp^t\geq q, x_i\in G\}$ and $[x_1,\cdots,x_s]$ is the iterated commutator $[\cdots[x_1,x_2],\cdots,x_s]$.

Is it true that the group ring of the completion ${\mathbb{Z}}/p[G^p]$ is the completion of the group ring ${\mathbb{Z}}/p[G]$ with respect to the products of the augmentation ideal? i.e. ${\mathbb{Z}}/p[G^p]\cong \varprojlim_{q}{\mathbb{Z}}/p[G]/I^q$, where $I$ is the augmentation ideal of the group ring ${\mathbb{Z}}/p[G]$?