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 ${\rm{(P)SL}}_2(q)$. 
Alexandre Borovik, \c{S}\"ukr\"u Yal\c{c}{\i}nkaya
math.GR