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The question is inspired by the following observation:
Let $p: X'\to X$ be a connected covering space where both spaces are suitably nice (say they are CW complexes), then $p: X' \to X$ is a principal bundle in at most one way: If the group of deck transformations $\mathrm{Deck}_p$ acts transitively on any fibre, then $p : X' \to X$ is a principal $\mathrm{Deck}_p$-bundle.
This is not true for general fibrations. Take for example any discrete space $D$ and its projection onto a point, then if $D$ has more than one element, $D \to *$ can be given more than one structure of a principal bundle.
My question is then:

Given a fibration $p: E \to B$, what are the minimal conditions needed for $p: E \to B$ to be a principal $\mathrm{Aut}_p$-fibration.

Here $\mathrm{Aut}_p$ should be interpreted as a suitable homotopical notion of a group (so presumably a grouplike $A_\infty/E_1$-space) consisting of all weak homotopy equivalences of $E$ which are compatible with the map $p$. Also I'm not particularly interested in point-set topological conditions; the conditions should hold for any model of homotopy types.

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