Let $q=p^f$ be a prime power. Is $P\Gamma L_2(q)$, the automorphism group of $PSL_2(q)$, a semidirect product of $PSL_2(q)$ by its outer automorphism group $Z_{\gcd(2,q-1)}\times Z_f$? If it is not in general, then for which $q$ this holds?
2026-03-28 03:01:08.1774666868
The extension of $PSL_2(q)$ by its outer automorphism group
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The answer is no for $q=9$, and yes when either $q$ is even, or when $f$ is odd. I am 99% confident that the extension is non-split when $q$ is odd and $f$ is even, but I am not sure how to go about proving that. I have just done some quick computer calculations, and I can confirm that it is non-split for $q= 9,25,49,81,121,625,729$.