In [4] we have studied the surjectivity of the forgetful homomorphism f(G, X) : K_G(X)→ K(X). The homomorphism gives informations about lifting actions on stabl vector bundles. One of the purpose of this paper is to study lifting actions on vector bundles and give actions explicitly for geometrical uses, for example, equivariant Hopf constructions and a lifting problem for other spaces than the spheres.
In section I we shall give a criterion for the existence of lifting actions which is obtained by G. Bredon's work [2]. Section 2 consists of results obtained by J. Folkinan's theorems [3], and Proposition 3 in [5]. Moreover we shall prove the equivariance for representatives of of generators of the groups _<π3>(SO(4)) and _<π7>(SO(8)). In section 3 we shall prove the equivariance of Bott maps [1], which present us various constructions of equivariant maps. In the last section we shall apply results in preceding sections and obtain a non existence theorem, equivanant Hopf constructions and a lifting property on complex plane bundles over the complex projective plane.