Let $G$ be a finite group and $K$ a field with field extension $L$ ($K$ perfect and $L$ finite field extension first for simplicity),
Let $S$ be a simple $KG$ module.
Viewed as a $LG$-module $S$ decomposes as a direct sum of simple $LG$-modules $L\_l$.



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> Question 1: Let $i \geq 1$ be fixed. Do we have $Ext\_{KG}^i(S,S) \neq 0$ if and only if $Ext\_{LG}^i(L\_l,L\_l) \neq 0$?
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The $L\_i$ should be Galois conjugates of each other so we should have $Ext\_{LG}^i(L\_l,L\_l) \neq 0$ if and only if $Ext\_{LG}^i(L\_t,L\_t) \neq 0$ for some $l \neq t$.



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> Question 2: Does this property also hold for arbitrary (possibly infinite) field extensions?
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Extensions for simple modules over group algebras
Let $G$ be the alternating group $A\_4$, let $K=\mathbb{F}\_2$, and $L=\mathbb{F}\_4$. Then there are two non-trivial one dimensional simple $LG$-modules, say $S\_1$ and $S\_2$. These lie over a single two dimensional $KG$-module $S$, so extending the field on $S$ gives $S\_1\oplus S\_2$. Let $i=1$. Neither $S\_1$ nor $S\_2$ extends itself, but $S$ does because $S\_1$ and $S\_2$ extend each other. So the answer to question 1 is no.


