Simple algebras and invariants of linear actions

I shall discuss the following topics:
(1) Given an algebraic group $G$, let $V$ be a finite-dimensional algebraic $G$-module,
which admits a structure of a simple (not necessarily associative) algebra $A$ such that $G=Aut (A)$. Then $V$ admits a close approximation to the analogue of classical invariant theory.
(2) What are the groups $G$ for which such a $V$ exists?
(3) Given $G$, what are the $G$-modules $V$ for which (1) holds?