  We investigate pairs (G,Y ), where G is a reductive algebraic group and Y a
purely-odd G-superscheme, asking when a pair corresponds to a quasi-reductive
algebraic supergroup G, that is, Gevis isomorphic to G, and the quotient
G/Gevis G-equivariantly isomorphic to Y . We prove that, if Y satisfies certain
conditions (basic quasi-reductive root data), then the question has a positive
answer given by an existence and uniqueness theorem. We call the corresponding
supergroups basic quasi-reductive. We then classify connected quasi-reductive
algebraic supergroups of monodromy type provided that: (i) the root system does
not contain 0; (ii) g := Lie(G) admits a non-degenerate even symmetric bilinear
form.
