Principal bundles in supersymmetry

We discuss rather basic problems on super principal bundles for $C^{\mathrm{\infty }}$-supermanifolds (over $\mathbb{R}$) and analytic supermanifolds over a complete field of arbitrary characteristic. Sharper answers are obtained for $C^{\mathrm{\infty }}$-supermanifolds and for homogeneous supermanifolds. Among others, it is proved for $C^{\mathrm{\infty }}$-supermanifolds that given a $C^{\mathrm{\infty }}$-supermanifold $B$ and a super Lie group $G$, there is a natural one-to-one correspondence (up to isomorphism) between the $G$-super principal bundles over $B$ and the $G_{\mathrm{r}\mathrm{d}}$-principal bundles over $B_{\mathrm{r}\mathrm{d}}$, where $G_{\mathrm{r}\mathrm{d}}$ (resp., $B_{\mathrm{r}\mathrm{d}}$) denotes the ordinary Lie group (resp., manifold) associated with $G$ (resp., $B$). To prove this, we show a generalized, equivariant version of the renowned Batchelor Theorem.