By 'work' I would like the correspondence between fiber functors (to finitely generated projective modules) and algebraic groups to be the same as in the field case.

Specifically, if $A$ is an affine ring, and if $\operatorname{Proj}(A)$ is the category of finitely generated projective A-modules, when can we say that a fiber functor $w:\mathcal{T}\to\operatorname{Proj}(A)$ corresponds to an algebraic group over $A$, where $\mathcal{T}$ is an $A$-linear tensor category.