Invariants of coverings of Serre bundles

The category of coverings of a given Serre bundle is investigated. By a covering mapping of one such bundle onto another we mean a morphism in the category of bundles consisting of covering mappings of total spaces and bases. An invariant of a covering from the indicated category is constructed, which is the conjugacy class of some subsequence of the homotopy sequence of the base bundle. Criteria for the existence of morphisms and isomorphisms for given coverings are obtained.