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?
The extension of $PSL_2(q)$ by its outer automorphism group
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group-theory
finite-groups
simple-groups