Denote the symmetric group of order $n!$ by $S_n$. Let $H:=S_p$ for an odd prime $p$.
Every finite field $k$ is a splitting field for $kH$, in particular $k:=\mathbb{F}_p$.
Questions:
Is $k:=\mathbb{F}_p$ also a splitting field for $kG$ where
a) $G:=H\times H$ ?
b) $G:=H \wr C_2$ ?
I would be interested in references appearing in the literature which deal with these (similar) questions.
Thank you in advance for the help.
Edit: $k$ is a splitting field of $S_n$ if the $k$-algebra $kS_n$ splits over $k$, i.e. if for every simple $kS_n$-module $M$, we have End$_{kSn}(M)\cong k$. (cf. Splitting fields of symmetric groups)
Remark: I looked at https://ncatlab.org/nlab/show/direct+product+group and due to remark 2.2 my questions might not have an affirmative answer for arbitrary groups, but I was interested, if the statement is nevertheless true in these special cases.