$\newcommand{\i}{\iota}$
The general notion that I am trying to disprove is that if we are given a fibration $X \to Y$ with fiber $F$ such that the delooping $BF$ exists, that there is a map $Y \to BF$ such that $F \to X \to Y \to BF$ is a fibration sequence.


This question came up when I was trying to see what went wrong when trying to define postnikov towers for nonsimply connected spaces.


Here is my proposed counterexample:


Let $X$ be a nonsimply connected space with the action of $\pi\_1(X) \curvearrowright \pi\_2(X)$ nontrivial. Since the meaning of a postnikov tower here is ambiguous let me explicitly define $X\_1=K(\pi\_1(X),1)$ and $X\_2$ the space obtained by killing all homotopy groups above dimension $2$, and let $X\_2 \to X\_1$ be the map obtained by just adding cells to $X\_2$ to kill homotopy at dimension 2.


I want to show that there is no map $X\_1 \to K(\pi\_2(X),3)=BK(\pi\_2(X),2)$ such that $K( \pi\_2(X),2) \to X\_2 \to X\_1 \to K(\pi\_2(X),3)$ is a fibration sequence.


Intuitively, the only candidate for such a map would be an element of $[X\_1, K(\pi\_2(X),3)]=H^3(X\_1, \pi\_2(X))$ arising as the transgression of the fundamental class of $K(\pi\_2(X),2)$ in the Serre Spectral Sequence of fibration $K( \pi\_2(X),2) \hookrightarrow X\_2 \to X\_1$. 


In order for this to make sense the fundamental class $\i$ of $K(\pi\_2(X),2)$ needs to be in the domain of the transgression, and it is not:


We can view $\i =Id \in Hom(\pi\_2(X), \pi\_2(X))$, $E\_2^{0,2}=Hom(\pi\_2(X), \pi\_2(X))^{\pi\_1(X)}$ where $\pi\_1(X)$ acts on the first $\pi\_2(X)$, and the transgression is $d\_3$, which in view of the vanishing of $E\_2^{1,1}$, acts on all of $E\_2^{0,2}$. 


Then we have $\iota \in Hom(\pi\_2(X), \pi\_2(X))$ is not $\pi\_1(X)$ invariant, and so it is not in the domain of the transgression.


This leaves the following question on how I can make this argument rigorous: 



> 
> How can I show that any candidate map $X\_1 \to K(\pi\_2(X),3)$ would have to arise as the transgression of the fundamental class of $K(\pi\_2(X),2)$?
> 
> 
> 



The inability to continue a fibration sequence even when a delooping exists
I haven't thought about the example that you describe, but here is a different kind of example. 


Consider a topological group $F$, and let $M$ denote the monoid of weak equivalences $F\to F$, so there is an evident inclusion $F\to M$. Let $X$ be any based space, and let $i\colon X\to CX$ be the inclusion of the base of the cone. Now suppose we have a based map $f\colon X\to M$. Take two copies of $CX\times F$ and identify $(i(x),u)$ in the first copy with $(i(x),f(x)(u))$ in the second. This gives a space $E(f)$ with evident maps $F\xrightarrow{j}E(f)\xrightarrow{q}\Sigma X$. I think that this is always a fibration up to homotopy. If the map $f$ does not factor (up to homotopy) through $F$, then we have a fibration that is not classified by a map $\Sigma X\to BF$.


Note here that $f$ is supposed to be a based map when we take the identity as the basepoint in $f$. One way to define such an $f$ is to take a based map $m\colon X\wedge F\to F$, and put $f(x)(u)=m(x\wedge u).u$.


For a minimal example of this, take $F=S^3=SU(2)$ and $X=S^1$. Let $\eta\colon S^3\to S^2$ be the Hopf map, and take $m=\Sigma\eta\colon X\wedge F\to F$. The resulting map $f\colon X\to M$ is nontrivial in homotopy, but $[X,F]=\pi\_1(S^3)=0$.


