Let \({\overline{spt2}}(n) \) denote the number of smallest parts in the overpartitions of n where the smallest part is not overlined and the smallest part is even. In recent years, congruence properties for certain SPT functions including \({\overline{spt2}}(n) \) have attracted the attention of many mathematicians. Motivated by their works, in this paper, we give characterizations of congruences modulo 2 and 4 for \({\overline{spt2}}(n) \) by using an identity involving the second order mock theta function B(q) due to Gu and Su, and the generating function for \({\overline{M2}}(r,4,n)\) which denotes the number of overpartitions of n where the second residual crank is congruent to r modulo 4.
