Let C_n denote the representation category of a finite supergroup generated
by purely odd n-dimensional vector space. We compute the Brauer-Picard group
BrPic(C_n) of C_n. This is done by identifying BrPic(C_n) with the group of
braided tensor autoequivalences of the Drinfeld center of C_n and studying the
action of the latter group on the categorical Lagrangian Grassmannian of C_n.
We show that this action corresponds to the action of a projective symplectic
group on a classical Lagrangian Grassmannian.