Lie supergroups supported over GL2GL2 and U2U2 associated to the adjoint representation

The algebraic classification of Lie superalgebras based on gl2 whose odd module is gl2 itself under the adjoint representation, together with the existence and uniqueness theorem for ODE’s in supermanifolds, are hereby used as in Lie’s theory to produce nonisomorphic Lie supergroups, all of which are supported over the same underlying Lie group GL2GL2, and whose Lie superalgebras of left-invariant supervector fields are isomorphic to those given by the algebraic classification. Their compact real forms are also studied so as to produce nonisomorphic Lie supergroups supported over the unitary group U2U2, and their corresponding maximal tori are described.