Automorphisms and real structures for a $\Pi$-symmetric super-Grassmannian

Any complex-analytic vector bundle $\mathbb E$ admits naturally defined homotheties $\phi_{\alpha}$, $\alpha\in \mathbb C^*$, i.e. $\phi_{\alpha}$ is the multiplication of a local section by a complex number $\alpha$. We investigate the question when such automorphisms can be lifted to a non-split supermanifold corresponding to $\mathbb E$. Further, we compute the automorphism supergroup of a $\Pi$-symmetric super-Grassmannian $\Pi\!\operatorname{Gr}_{n,k}$, and, using Galois cohomology, we classify the real structures on $\Pi\!\operatorname{Gr}_{n,k}$ and compute the corresponding supermanifolds of real points.