One way of constructing coalgebra objects in the homotopy theorist's category of spectra is to take the suspension spectrum of a space, with the diagonal providing a cocommutative comultiplication. If, moreover, we assume that the space (call it $X$) of interest has some suitably coherent multiplication on it (e.g. it's an $n$-fold loop space for some $n>0$), we can say that $\Sigma_+^\infty X$ is a bialgebra, with an $E_n$-multiplication and an $E_\infty$-comultiplication (whatever that means...).

In discrete algebra, one can construct quotient Hopf-algebras. In other words, given a Hopf-algebra $H$ and a normal sub-Hopf-algebra $B\subset H$, one can construct $H//B=H/(HB_+)$ where $B_+$ is the part of $B$ away from the augmentation ideal. One then can reconstruct $B$ just by looking at $H//B$ and $H$ by using the cotensor product. Or, in other words, the primitives of $H$ under the induced $H//B$ coaction: $H\to H\otimes H\to H\otimes H//B$, are precisely $B$.

My question is, when can one do the same thing for spectral bialgebras or Hopf-algebras? Clearly one must have a more "homotopical" notion of cotensor. In particular, one might notice that the usual diagram defining the cotensor product is simply the bottom two entries in a cosimplicial diagram, and the cotensor product of two spectral comodules can be defined as the limit of a cobar-type construction. But we can assume that we've got a decent notion for that.

In particular, given a fibration of spaces, $F\to E\to B$, one can produce a fibration of topological associative groups $\Omega F\to \Omega E\to \Omega B$ and then take suspension spectra to obtain a map of spectral Hopf-algebras (I believe), $\Sigma^\infty_+\Omega F\to \Sigma^\infty_+\Omega E\to \Sigma^\infty_+\Omega B$. Is this something we can think of as analogous to $B\to H\to H//B$ above? Perhaps we need to put stronger conditions in place to make it so? Moreover, it would seem that I can recover $\Sigma^\infty_+\Omega F$ from a cobar construction involving $\Sigma^\infty_+\Omega E$ and $\Sigma^\infty_+\Omega B$. Is that far off the mark? This is ultimately a part of my attempt to answer my previous related question.

Moreover, one can then ask the following question: we have a change of rings isomorphism (in discrete algebra) $$Ext_H(M,B\otimes N)\cong Ext_{H//B}(M,N).$$ Can this be realized as an equivalence of homotopy limits: $$C(X,\Sigma^\infty_+\Omega E,\Sigma^\infty_+ \Omega B)\simeq C(X,\Sigma^\infty_+\Omega F,\mathbb{S})$$ where $X$ is a $\Sigma^\infty_+E$-comodule?