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.