A generalisation of orbit algebras

We introduce generalised orbit algebras. The purpose here is to measure how some combinatorial properties can characterize the action of a group of permutations of the elements of the set $\groundset$ on the subsets of $\groundset$, that is to say, to try to characterize orbit algebras. A nontrivial example solving a problem of Pouzet and Rosenberg is provided to show that the two notions don't coincide.