Construction of some subgroups in black box groups ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(q)$ and ${(\mathrm{P})\mathrm{S}\mathrm{L}}_2(q)$

For the black box groups $X$ encrypting ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(q)$, $q$ odd, we propose an algorithm constructing a subgroup encrypting ${\mathrm{S}\mathrm{y}\mathrm{m}}_4$ and subfield subgroups of $X$. We also present the analogous algorithms for black box groups encrypting ${(\mathrm{P})\mathrm{S}\mathrm{L}}_2(q)$.