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For each finite primitive subgroup $G$ of $\operatorname{PGL}_4(\mathbb{C})$,we find all the smooth $G$-invariant quartic surfaces. We also find all thefaithful representations in $\operatorname{PGL}_4(\mathbb{C})$ of the smoothquartic $G$-invariant surfaces by the groups: $\mathfrak{A}_5,\mathfrak{S}_5,\operatorname{PSL_2(\mathbb{F}_7)},\mathfrak{A}_6,\mathbb{Z}_2^4\rtimes\mathbb{Z}_5$and $\mathbb{Z}_2^4\rtimes D_{10}$. The primitive representation of thesegroups are precisely the subgroups of $\operatorname{PGL}_4(\mathbb{C})$ forwhich $\mathbb{P}^3$ is not $G$-super rigid. As a byproduct, we show that thesmooth quartic surface with the biggest group of projective automorphism isgiven by $\{ x_0^4 + x_1^4 + x_2^4 + x_3^4 + 12 x_0 x_1 x_2 x_3= 0\}$ (uniqueup to projective equivalence).

Invariant Smooth Quartic Surfaces by all Finite Primitive Groups of $\operatorname{PGL}_4(\mathbb{C})$