Let $KG$ be a group algebra of a finite group $G$ such that the characteristic of $K$ divides the group order.

Question: When is a block of a group algebra (or even the whole group aglebra) a Koszul algebra? Is there a classification?

For example in characteristic two, the group algebra $KG$ is isomorphic to $K[x]/(x^2)$ when $G$ has two elements and here the group algebra is a Koszul algebra with Koszul dual the polynomial ring $K[x]$.