Edit: According to essential comment of YCore I revise the question.
Let $A$ be a finite dimensional graded algebra which is a unital, super commutative and associative algebra. Is there a Lie group $G$ whose differential graded algebra of all $G$-left invariant differential forms be isomorphic to $A$? This is a differential form analogy of the classical fact that " Every finite dimensional Lie algebra is the Lie algebra of a Lie group".
I browsed this arXiv paper but I did not find a result for the graded algebra case.