Let p:E−>B be a principal fibration with classifying map w:B−>C. It
is well-known that the group [X,ΩC] 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,ΩB]. 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,ΩC].
As an example, [X,S2] is enumerated for X a 4-complex.Comment: 12 page