Consider a group $G$ acting on a set $Ω$, the vector $v_{a,b}$ is a vector with the entries indexed by the elements of $G$, and the $g$-entry is 1 if $g$ maps $a$ to $b$, and zero otherwise. A $(G,Ω)$-Cameron-Liebler set is a subset of $G$, whose indicator function is a linear combination of elements in $\{v_{a, b}\ :\ a, b \in Ω\}$. We investigate Cameron-Liebler sets in permutation groups, with a focus on constructions of Cameron-Liebler sets for 2-transitive groups.