We know that the $\mathbb{Z}/2$ central extension of $ SO(d) $ can give a nontrivial double/universal cover of $SO(d)$ known as the $Spin(d)$ group. They have this relation $$ 1 \to \mathbb{Z}/2 \to Spin(d) \to SO(d) \to 1.$$
My puzzle is that
Is there a Lie Supergroup for the Lie group $SO(d)$? What it will be? This definition is too general https://ncatlab.org/nlab/show/supergroup and I hope to look for better answers here.
how is the $Spin(d)$ relates to the Supergroup for SO group?
What is the relation $Spin(d)$ v.s. Supergroup for $SO(d)$?