The Lang-Trotter conjecture on average for genus-$2$ curves with $S_3$ reduced automorphism group

For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjectured that the number of primes $p <X$ at which $E$ has a supersingular reduction is asymptotically equal to $c\sqrt{X}/\log X$, where $c>0$ is a constant depending only on $E$. While it remains an open question, an average estimation related to the Lang-Trotter conjecture was established by Fouvry and Murty. This result is called the Lang-Trotter conjecture on average. We extend the Lang-Trotter conjecture to curves of genus $2$ and obtain a similar result to the Lang-Trotter conjecture on average for the family of curves $C_{\lambda}:y^2=x(x-1)(x-{\lambda})(x-(\lambda-1)/{\lambda})(x-1/ (1-\lambda))$. These curves are characterized as curves of genus $2$ with reduced automorphism group containing symmetric group $S_3$.