On the Symmetry between Lie group Lie groupoid action

The notion of an action of a differentiable groupoid (G, B) generalize the action of a lie group. G relative to a differentiable bundle (E, P, B)Our work here is to give a special type of differentiable groupoid (lie groupoid) (G, B) in order to make (E, P, B) a principal fiber bundle G(E, P, B) by which we get a symmetry between principal action of a lie group G and that of a lie groupoid G and then a symmetry (conjugation form) between their principal action morphisms T ,T * respectively.