In this paper we characterize the rank two vector bundles on P 2 which are invariant under the action of Gp := Stabp(PGL(3)), that fixes a point in the projective plane, GL := StabL(PGL(3)), that fixes a line, and T = Gp ∩ GL. Moreover, we prove that the geometrical configuration of the jumping locus induced by the invariance does not, on the other hand, characterize the invariance itself. Indeed, we find infinity families that are almost uniform but not almost homogeneous.