Let $G$ be a reasonably nice group, say residually finite if need be.

We may consider the group algebra $\mathbb{Z}[G]$.

Let $\widehat{\mathbb{Z}[G]} := \varprojlim_I\mathbb{Z}[G]/I$ be the profinite completion, where $I$ runs over all ideals of finite index.

Let $\widehat{\mathbb{Z}}[[\widehat{G}]] := \varprojlim_{n,U}(\mathbb{Z}/n)[G/U]$ be the "completed group algebra", where $n$ runs over all positive integers, and $U$ all finite index normal subgroups of $G$.

When are these two objects the same? (Are they always the same? Perhaps they're the same when $G$ is abelian?)

This seems like a natural question, but I cannot find it addressed anywhere, and I can't seem to deduce anything nontrivial in either direction.