Group Actions · Hard problem

$S_3$ acts on the $8$ functions $f:\{1,2,3\}\to\{A,B\}$ by $(\sigma\cdot f)(i)=f(\sigma^{-1}(i))$ (relabeling the domain). Let $f_0$ satisfy $f_0(1)=A,f_0(2)=A,f_0(3)=B$. Find $|\operatorname{Stab}(f_0)|$.