There's a sheafy way to write down the $\infty$-category of super-$\infty$-groupoids, as detailed on the nlab. Can this notion be reformulated by saying that a super-$\infty$-groupoid is an ordinary $\infty$-groupoid plus some homotopy-theoretic data?
The sheafy notion is this: let $sMan$ be the usual category of supermanifolds, regarded as a site with the usual Grothendieck topology (a cover is a jointly surjective collection of open embeddings). Consider the $\infty$-category $Sh(sMan)$ of sheaves of $\infty$-groupoids on this site, and then localize at the morphisms $\mathbb R^{p|q} \to \mathbb R^{0|q}$. If we did this same process with the site $Man$ of ordinary manifolds, we'd end up with the $\infty$-category of $\infty$-groupoids, so when we do it here, the resulting $\infty$-category deserves to be called the $\infty$-category $sGpd_\infty$ of super-$\infty$-groupoids. There is a natural forgetful functor $|-|: sGpd_\infty \to Gpd_\infty$ induced by the forgetful functor $sMan \to Man$.
Question: What additional homotopy-theoretic data is needed to recover a super-$\infty$-groupoid $X$ from its underlying $\infty$-groupoid $|X|$? Can this be formulated cleanly enough to give an alternative construction of the $\infty$-category $sGpd_\infty$?
The kind of data I have in mind is that maybe $X$ can be recovered from the data of $|X|$ plus maybe a map from $|X|$ into the classifying space of some sort of structure group, or something of that sort.
