Coverings of Steenrod bundles

We study a covering in the category of Serre bundles. It is assumed that the base of the covering is a Steenrod bundle (fibre bundle) with a connected structure group. We obtain a condition under which the covering bundle has the same property, and the covering maps of bases, total spaces, fibers, and structure groups form a morphism in the category of Steenrod bundles. It is proved that in the case when the base of the covering is a principal bundle, this condition is always satisfied, and therefore the covering bundle is also principal.