  Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It
is well-known that the group $[X,\Omega C]$ acts on $[X,E]$ with orbit space
the image of $p_#$, where $p_#: [X,E] -> [X,B]$. The isotropy subgroup of the
map of $X$ to the base point of $E$ is also well-known to be the image of $[X,
\Omega B]$. The isotropy subgroups for other maps $e:X -> E$ can definitely
change as $e$ does.
  The set of homotopy classes of lifts of $f$ to the free loop space on $B$ is
a group. If $f$ has a lift to $E$, the set $p_#^{-1}(f)$ is identified with the
cokernel of a natural homomorphism from this group of lifts to $[X, \Omega C]$.
  As an example, $[X,S^2]$ is enumerated for $X$ a 4-complex.
