Assume $X$, $E$ and $G$ are topological groups and $1\to X\to E\to G\to 1$ a short exact sequence of continuous group homomorphisms.
Under which of these conditions is $E$ a profinite group?


(i) $G$ profinite, $X$ finite


(ii) $G$ profinite, $X$ pro-$p$ for some prime $p$


(iii) $G$ profinite, $X$ profinite


In those case where $E$ is profinite, is there a way to write down the inverse system for $E$ explicitly in terms of the inverse system of $G$ (and $X$)


Thank you very much.



Are extensions of profinite groups profinite?
The answer to all three questions is no in general. You need to assume in addition that $G$ carries the quotient topology from $E$. Otherwise, starting from any such exact sequence with $E$ compact and $G$ infinite, you can endow the compact group $E$ with the topology inherited from the embedding into $E\times G\_d$ where $G\_d$ is $G$ with the discrete topology; let $E'$ be the resulting group; then $1\to X\to E'\to G\to 1$ is an exact sequence of continuous group homomorphisms, $X$ and $G$ are compact but $E'$ is not. Here $X$ can even be the trivial group, so $E'=G\_d$.


