When is the group algebra a product of local rings up to Morita equivalence?

Question

Let $KG$ be the group algebra of a finite group $G$ over a field of characteristic $p$.

Question 1: Is there a characterisation when $KG$ is Morita equivalent to a product of local rings?

This should be true for example when $G$ has a normal $p$-complement.

Question 2: Is there a description of the group algebra when $G$ has order $p^n q$ for primes $p<q$ in terms of its $p$-Sylowsubgroup $P$ (if it helps, assume $p=2$)? Maybe even quiver and relations of $G$ are known in terms of quiver and relations of $KP$?

Answer 1

At least if the field is algebraically closed (or sufficiently large), a finite group has a normal $p$-complement if and only if its principal block is local. For example, this is Corollary 6.13 of

Navarro, Gabriel, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series. 250. Cambridge: Cambridge University Press. x, 287 p. (1998). ZBL0903.20004.