Simple algebras and invariants of linear actions

I shall first describe a general construction that yields, for every finite dimensional $G$-module $V$ of a group $G$ admitting a structure of simple (not necessarily associative) algebra $A$ such that $\operatorname{Aut} A=G$, some polynomial $G$-invariant functions on the direct sum of several copies of $V$.
I shall then address the following three arising questions:

• (1) How many functions are obtained in this manner? In particular, do they generate the field of all $G$-invariant rational functions?
• (2) When does such a structure of simple algebra exists?
• (3) Which groups arise in this context?