Let $G$ be any group, and $\widehat{G}$ its profinite completion. Is it true that $\widehat{\widehat{G}}=\widehat{G}$, i.e. is it true that $\widehat{G}$ is (canonically isomorphic to) its own profinite completion? It seems that it should follow from the universal property of the profinite completion, but I don't see how.
Thanks in advance for any solutions or suggestions.
I think there are counterexamples concerning the iterated profinite completions of groups, such as the abosulte Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, which as a profinite group does not equal its own profinite completion (because it has subgroups of finite index that are not open). A deep theorem of Segal and Nikolov from $2007$ (using the classification of finite simple groups) implies that if $\widehat{G}$ is topologically of finite type then $\widehat{G}$ and $\widehat{\widehat{G}}$ are isomorphic.
Edit: Counterexamples to $\widehat{\widehat{G}}\simeq \widehat{G}$ can be found here (section 5, first line) arxiv.org/pdf/0801.2955.