$\newcommand{\i}{\iota}$ 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 $K(\pi_2(X),2)$?