In this talk we give a refinement of the seminal work of Bhargava and Shankar on enumerating $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of binary quartic forms with bounded invariants. In particular, we show how to count such equivalence classes of irreducible quartic forms with Galois groups not isomorphic to the symmetric group $S_4$ nor the alternating group $A_4$. This is joint work with Cindy Tsang.