In this paper, we develop a simple method for computing the stabilizer subgroup of a subgroup of $$D(g)={{alpha}{in}mathbb{F}_q|there;is;a;{eta}{in}{mathbb{F}}^x_q;such;that;{eta}^n=g(alpha)}$$ in $PSL_2(mathbb{F}_q)$, where q is a large odd prime power, n is a positive integer dividing q-1, and $g(x){in}mathbb{F}_q[x]$. As an application, we construct new infinite families of 3-designs (cf. Examples 3.4 and 3.5).