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We consider a $J_b(\mathbb{Q}_p)$-torsor on the supersingular locus of theSiegel threefold constructed by Caraiani-Scholze, and show that it induces anisomorphism between a free group on a finite number of generators, and thegroup of self-quasi-isogenies of a supersingular abelian surface, respecting aprincipal polarization and a prime-to-$p$ level structure. Along the way, weclassify certain pro-\'etale torsors in terms of the pro-\'etale fundamentalgroup, describe the category of geometric covers of non-normal schemes, and usethis to compute pro-\'etale fundamental groups of curves.

Quasi-isogeny groups of supersingular abelian surfaces via pro-\'etale fundamental groups