  We study fiber bundles where the fibers are not a group $G$, but a free
$G$-space with disjoint orbits. These bundles closely resemble principal
bundles, hence we call them semi-principal bundles. The study of such bundles
is facilitated by defining the notion of a basis of a $G$-set, in analogy with
a basis of a vector space. The symmetry group of these bases is a wreath
product. Similar to vector bundles, using the notion of a basis induces a frame
bundle construction, which in this case results in a principal bundle with the
wreath product as structure group. This construction can be formalized in the
language of a functor, which retracts the semi-principal bundles to the
principal bundles. In addition, semi-principal bundles support parallel
transport just like principal bundles, and this carries over to the frame
bundle.
