  Algebraic models for equivariant rational homotopy theory were developed by
Triantafillou and Scull for finite group actions and $S^1$ action,
respectively. They showed that given a diagram of rational cohomology algebras
from the orbit category of a group $G$, there is a unique minimal system of
DGAs and hence a unique equivariant rational homotopy type that is weakly
equivalent to it. However, there can be several equivariant rational homotopy
types with the same system of cohomology algebras. Halperin, Stasheff, and
others studied the problem of classifying rational homotopy types up to
cohomology in the non-equivariant case. In this article, we consider this
question in the equivariant case. We prove that when $\mathbb{Z}_p$ under
suitable conditions, the equivariant rational homotopy types with isomorphic
cohomology can be reduced to the non-equivariant case.
