The Lang-Trotter conjecture on average for genus-$2$ curves with Klein-$4$ reduced automorphism group

For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjecture $$\mathrm{\#}\{p<X\mid E\text{ has a supersingular reduction at }p\}\sim \frac{c\sqrt{X}}{\log X}$$ as $X\to \mathrm{\infty }$, where $c>0$ is a constant depending only on $E$. Fourvy and Murty obtained an average estimation related to the Lang-Trotter conjecture, called the Lang-Trotter conjecture on average. We considered the Lang-Trotter conjecture for curves of genus 2, and obtained a similar result to the Lang-Trotter conjecture on average for the family of curves $C_{\lambda }:y^2=x(x-1)(x+1)(x-\lambda )(x-1\mathrm{/}\lambda )$. Such curves are characterized as curves of genus two with reduced automorphism group containing the Klein $4$-group.