TITLE: Delooping a fibration sequence with loopspace fiber and finite CW complexes
QUESTION [5 upvotes]: The following question is somewhat similar to a previous one on MathOverflow, except that my application does not directly involve Eilenberg-MacLane spaces $K(G,n)$, and so I don't see the immediate need for $n$-simple maps as in the theory of Postnikov $k$-invariants.
Suppose that $\Omega T \longrightarrow F \longrightarrow E$ is a homotopy fibration sequence, where $T$ and $F$ are connected finite CW complexes.  Is there a quantification in some sort of obstruction theory for extending this sequence further to the right in the form $F \longrightarrow E \longrightarrow T$ ?  This is a 1950s-era topology question.
Of course, the original sequence extends further to the right as $F \longrightarrow E \longrightarrow B hAut(\Omega T)$.  First, this is by Stasheff's paper for finite CW complexes as the fiber and any CW base, working with products in the full topological category [Sta63]. Later, this is by May's simplicial upgrade to infinite CW complexes as the fiber, working in the compactly generated category [9.5, 9.8][May75].
In my application, $T$ happens to have the structure of a Lie group, but this is a happy accident of its dimension, and I'm hoping for an answer that can work independent of this fact.
[Sta63]: James D Stasheff, A classification theorem for fibre spaces, Topology 2, 1963
[May75]: J Peter May, Classifying spaces and fibrations, Memoirs AMS 1:155, 1975
EDIT: As suggested, I renamed the base space $B$ to $T$, in order to not confuse it with the classifying space functor.  Also, I added two citations.

REPLY [6 votes]: This question is addressed in the paper 
Ganea, T., Induced fibrations and cofibrations, Trans. Am. Math. Soc. 127, 442-459 (1967). ZBL0149.40901.
A first observation is that $\Omega T\to F\to E$ extends to the right if and only if it is induced from the based path fibration $\Omega T\to PT\to T$ by a map $p:E\to T$. In Section 2, various sufficient conditions for such a fibration to be induced are given.
Two sample results:
Corollary 2.5: Suppose that $\pi_q(E)\neq 0$ only if $m\le q\le n + m-1$ and that
$\pi_q(\Omega T) \neq 0$ only if $n\le q\le n + m-1$, where $n\ge m\ge 2$. If the Whitehead product pairing $W:\pi_n(\Omega T) \otimes \pi_m(F)\to \pi_{n + m-1}(F)$ vanishes, then $\Omega T\to F\to E$ is induced.
Theorem 2.10: Suppose $\Omega T$, $F$ and $E$ all have the homotopy type of aspherical CW complexes. Then $\Omega T\to F\to E$ is induced if and only if the image of the induced map $\pi_1(\Omega T)\to \pi_1(F)$ lies in the center.