Extension of scalars in single-block group algebras

Question

The question in the title is a follow-up for this one. I am now specifically interested in simple groups, and the results of papers I found (by Harris, and Brockhaus and Michler) imply that

If $G$ is a finite non-Abelian simple group, and $\overline{\Bbb F_p} G$ only has a single block, then $p = 2$ and $G$ is either $M_{22}$ or $M_{24}$.

The case $p = 2$ is stated in Harris Theorem 4(a) in plain words, and for $p > 2$ it is (or seems to be...) a consequence of his Theorem 1(a). (The proof depends on the results of Brockhaus and Michler, who have done the Lie type case.)

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However all of this is proven over the algebraic closure of a finite field, rather than the finite field itself (which I am interested in). For semisimple group algebras, the splitting of a block in a field extension is commonplace, however the principal block is always 1-dimensional and thus can't split. Can it split in the modular setup above? In other words,

"Problem." Let $k = \Bbb F_{p^m}$, $K = \overline{k}$, $G$ a finite simple group. Prove that if $kG$ has only the principal block, then the same is true of $KG$.

My attempt (that does not use simplicity of $G$). By Lemma 13.2 in Blackburn and Huppert Finite groups II, p. 185, for an irreducible $FG$-module "being in the principal block" is equivalent to "being annihilated by $(\sum\limits_{g \in C} g) - |C|$ for all conjugacy classes $C$" (this Condition is independent of properties of $F$). We have $KG \cong K \otimes_k kG$, and presumably any irreducible $KG$-module $M$ is a constituent (Edit 14 Jan 2024: isomorphic to a submodule) of $K \otimes_k \mu$ for some irreducible $kG$-module $\mu$ (the most phony point — is this still true in the modular case?) Then if $M$ fell outside the principal block of $KG$, so would $\mu$ in $kG$ by the above Condition. "$\square$"

(In 12.6, the blocks are defined for arbitrary fields $F$ of positive characteristic.)

My questions: 1) Is this "argument" reasonable, especially the part in italics? How to prove it? 2) If it isn't, is the "problem" statement even true?

Answer 1

From the same book by Blackburn and Huppert, p. 21:

Theorem 1.18a). Let $L/K$ be a Galois extension and $V$ an irreducible $LG$-module. Then there exists an irreducible $KG$-module $W$ s.t. $V$ is isomorphic to a direct summand of $W \otimes_K L$. Furthermore, $W$ is unique up to isomorphism.

This is proved by reducing to the semisimple part.

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The argument can be made sound if one notes that every irreducible (or even just indecomposable) module lies in some block, and every block contains some irreducible module — see Definition 12.6.