Secondary Term of Asymptotic Distribution of $S_3\times A$ Extensions over $\mathbb{Q}$

We combine a sieve method together with good uniformity estimates to prove a secondary term for the asymptotic estimate of $S_3\times A$ extensions over $\mathbb{Q}$ when $A$ is an odd abelian group with minimal prime divisor greater than $5$. At the same time, we prove the existence of a power saving error when $A$ is any odd abelian group.