Construction of some subgroups in black box groups {rm {PGL}}₂(q) and {rm{(P)SL}}₂(q)

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