On extensions for Ree groups of type F_4

Let k be an algebraically closed field of characteristic p=2. Let G=F_4 be simply connected over k and let σ:G\to G be an endomorphism such that the fixed point set G(σ) is a Ree group. We show, using the methods of Bendel--Nakano--Pillen, that self-extensions of simple kG(σ)-modules vanish generically and that for all but the first few Ree groups of type F_4, the 1-cohomology for G(σ) with coefficients in simple kG-modules can be identified with the 1-cohomology for G with coefficients in (possibly different) simple G-modules.

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