Affine algebraic super-groups with integral

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We generalize to the super context, the known fact that if an affine algebraic group G over a commutative ring k acts freely (in an appropriate sense) on an affine scheme X over k, then the dur sheaf of G-orbits is an affine scheme in the following two cases: (I) G is finite; (II) k is a field, and G is linearly reductive. An emphasize is put on the more difficult generalization in the second case; the replaced assumption then is that an affine algebraic super-group G over an arbitrary field has an integral. Those super-groups which satisfy the assumption are characterized, and are seen to form a large class if Hopf-algebraic techniques including bosonization are applied to prove the results.