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For the black box groups $X$ encrypting ${\rm{PGL}}_2(q)$, $q$ odd, wepropose an algorithm constructing a subgroup encrypting ${\rm{Sym}}_4$ andsubfield subgroups of $X$. We also present the analogous algorithms for blackbox groups encrypting ${\rm{(P)SL}}_2(q)$.

Construction of some subgroups in black box groups ${\rm {PGL}}_2(q)$ and ${\rm{(P)SL}}_2(q)$