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 ?