Take a profinite group $G=\varprojlim G_\alpha$. We know that the inner automorphism group $\text{Inn}(G)$ of $G$ is profinite since $\text{Inn}(G)=G/Z(G)$, and the quotient of a profinite group by a closed normal subgroup (which $Z(G)$ must be as the intersection of centralizers) is profinite. I'm considering another object $\widehat{\text{Inn}}(G)=\varprojlim\text{Inn}(G_\alpha)$. Now I would like to show that $\text{Inn}(G)=\widehat{\text{Inn}}(G)$. A priori there may be things in $\widehat{\text{Inn}}(G)$ that are not in $\text{Inn}(G)$, so I believe that showing this is a worthwhile goal. There is an embedding $\text{Inn}(G)\hookrightarrow\widehat{\text{Inn}}(G)$ and I've managed to show that $\text{Inn}(G)$ is dense in $\widehat{\text{Inn}}(G)$. It remains to show that $\text{Inn}(G)$ is closed in $\widehat{\text{Inn}}(G)$. I believe that one way to go about this is to take a limit point $\varphi$ of $\text{Inn}(G)$ and show that any open subset of $\widehat{\text{Inn}}(G)$ that contains $\varphi$ must intersect $\text{Inn}(G)$. Can someone help me with this argument?
Another possible way involves considering what the basic closed sets look like in a Stone topological space presented as an inverse limit. I know what the basic open sets look like, but what do basic closed sets look like?
Edit: I want to assume all the maps $f:G_\alpha\to G_\beta$ are surjective.