Class numbers and congruences modulo 8 for a spt function on overpartitions

In 2014, Garvan and Jennings-Shaffer defined a function $\overline{{\rm spt}}(n)$ which denotes the number of smallest parts in the overpartitions of $n$ where the smallest part is not overlined. In recent years, characterizations of congruences modulo 2 and 4 for $\overline{\textrm{spt}}(n)$ were established. Garvan and Jennings-Shaffer proved that for $n\geq 1$, $\overline {{\rm spt}}( n)$ is odd if and only if $n $ is a square or twice a square. Recently, Yao gave a characterization of congruences modulo 4 for $\overline{{\rm spt}}( n)$. In this paper, we present a characterization of congruences involving class numbers modulo 8 for $\overline{{\rm spt}}( n)$ by using some identities on mock theta functions due to Gu and Su. In addition, we show that the arithmetic density of the set of integers such that $\overline{{\rm spt}}(n)\equiv 0 \pmod{8}$ is 1.