${\mathbb{A}}^{0\mathrm{\mid }1}$-torsors, quotients by free ${\mathbb{A}}^{0\mathrm{\mid }1}$-actions, and embeddings into $\mathrm{\Pi }$-projective spaces and super-Grassmannians $G(1\mathrm{\mid }1,n\mathrm{\mid }n)$

We study embeddability of superschemes into $\mathrm{\Pi }$-projective spaces and into supergrassmannians $G(1\mathrm{\mid }1,n\mathrm{\mid }n)$. We give some criteria based on the relation with ${\mathbb{A}}^{0\mathrm{\mid }1}$-torsors and ${\mathbb{A}}^{0\mathrm{\mid }1}$-fibrations. We also prove the existence of nice quotients for free actions of ${\mathbb{A}}^{0\mathrm{\mid }1}$ on superschemes.