Suppose that $X$ is a connected $E\_{\infty}$-space, naturally $\Omega X$ is also an $E\_{\infty}$-space. Can we classify all $E\_{\infty}$-extensions of $X$ by $\Omega X$ (up to homotopy). I mean the following: we would like to classify of homotopy fiber sequences $A\rightarrow B\rightarrow C$ where $A\sim \Omega X$ and $C\sim X$ as $E\_{\infty}$-spaces and $B\rightarrow C$ , $A\rightarrow B$ are maps of $E\_{\infty}$-spaces in particular we assume that $B$ is an $E\_{\infty}$-space. (space could mean a simplicial set or a topological space)


There are two obvious extensions of $E\_{\infty}$-space the first one is when $B\sim X\times \Omega X$ and the second one is when $B=PX$ the path space. Is there others ? My first guess was that the set of extensions is a group and more precisely the set of homotopy classes of maps of spectra $[X,X]$ where $X$ is seen as a connective spectra. Maybe I'm wrong but the first extension corresponds to the homotopy class of the trivial map $X\rightarrow \ast\rightarrow X$ and the second extension corresponds to the identity map $id: X\rightarrow X$. But there should be other extension in bijective correspondence with $[X,X]$. Is it correct ? 



Homotopy extension of $E_{\infty}$-spaces
This is probably belaboring the obvious, but just take seriously the equivalence between grouplike $E\_{\infty}$ spaces and connective spectra. See for example 


[Equivalence between $E\_\infty$-spaces and connective spectra](https://mathoverflow.net/questions/90379/equivalence-between-e-infty-spaces-and-connective-spectra?rq=1)


The asumption that X is connected means that the associated spectrum is connected and not just connective. So we may as well just ignore $E\_{\infty}$ spaces (much as I hate to do that!) and take $X$ to be a connected spectrum. One is asking for all fiber sequences $\Sigma^{-1}X \to Y \to X$ of spectra or equivalently all fiber sequences $Y \to X \to X$. That is, Y is equivalent to the fiber of a map $X\to X$. A quick triangulated category type argument shows that equivalence classes of such fibers correspond bijectively to maps $X\to X$, that is to elements of $[X,X]$.


