  We compute all signatures of $PSL_2(\mathbb{F}_7)$, and
$PSL_2(\mathbb{F}_{11})$ which classify all orientation preserving actions of
the groups $PSL_2(\mathbb{F}_7)$, and $PSL_2(\mathbb{F}_{11})$ on compact,
connected, orientable surfaces with orbifold genus $\geq 0$. This
classification is well-grounded in the other branches of Mathematics like
topology, smooth, and conformal geometry, algebraic categories, and it is also
directly related to the inverse Galois problem.
