Recall that a Nakayama algebra (also called serial algebras in the literature) is an algebra such that every indecomposable module has a unique composition series. In the book "Classical artinian rings and related topics" from 2009 by Yoshitomo Baba und Kiyoichi Oshiro one can find the following two questions at the end of the book:

Let G be a finite group and K be an algebraic closure of a field k. If kG is a Nakayama algebra, is KG a Nakayama algebra? And how is the converse?

For which algebraically closed fields $K$ and for which $n$ are the group algebras $KS_n$ and $KA_n$ Nakayama algebras, where $S_n$ is the symmetric and $A_n$ the alternating group?

(the original formulation for 2. is: "Let K be an algebraically closed field. Are there infinitely many $KS_n$ or $KA_n$ which are non-semisimple Nakayama algebras?")

Is an answer to those problems known now?

edit: It would be interesting to know whether there are quick computer programs available to test whether $KS_n$ or $KA_n$ are Nakayama algebras for $K$ being the algebraic closure of a prime field and given $n$.