Basic quasi-reductive root data and basic reductive supergroups

We investigate %the problem what pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd superscheme, asking when they correspond to a quasi-reductive algebraic supergroup $\mathbb{G}$ such that $\mathbb{G}_{\text{ev}}$ is isomorphic to $G$, and the quotient $\mathbb{G}\slash \mathbb{G}_{\text{ev}}$ is isomorphic to $Y$. Our result says that if $Y$ satisfies certain conditions of so-called basic quasi-reductive root data, then the question can be solved by the existence and uniqueness theorem (the corresponding supergroups are called basic quasi-reductive supergroups). Such basic quasi-reductive supergroups of monodromy type can be classified, all of which are reductive supergroups. Moreover, a connected quasi-reductive algebraic supergroup $\mathbb{G}$ must fall in the list of the above classification as long as it satisfies the condition as below (i) the root system does not contain $0$; (ii) $\mathfrak{g}:=\Lie(\mathbb{G})$ admits a non-degenerate even symmetric bilinear form.