2-complex symmetric composition operators on $H^2$

In this paper, we study 2-complex symmetric composition operators with the conjugation $J$ on the Hardy space $H^2$. More precisely, we obtain the necessary and sufficient condition for the composition operator $C_\phi$ to be 2-complex symmetric when the symbols $\phi$ is an automorphism of $\mathbb D$. We also characterize the 2-complex symmetric composition operator $C_\phi$ on the Hardy space $H^2$ when $\phi$ is a linear fractional self-map of $\mathbb D$.