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It was proved by the first-named author and Zubkov [13] that given an affinealgebraic supergroup $\mathbb{G}$ and a closed sub-supergroup $\mathbb{H}$ overan arbitrary field of characteristic $\ne 2$, the faisceau $\mathbb{G}\tilde{/} \mathbb{H}$ (in the fppf topology) is a superscheme, and is,therefore, the quotient superscheme $\mathbb{G}/\mathbb{H}$, which hasdesirable properties, in fact. We reprove this, by constructing directly thelatter superscheme $\mathbb{G}/\mathbb{H}$. Our proof describes explicitly thestructure sheaf of $\mathbb{G}/\mathbb{H}$, and reveals some new geometricfeatures of the quotient, that include one which was desired by Brundan [2],and is shown in general, here for the first time.

Geometric construction of quotients $G/H$ in supersymmetry