On Examples and Classification of Frobenius Objects in Rel

We give some new examples of Frobenius objects in the category of sets and relations $\mathbf{\text{Rel}}$. One example is a groupoid with a twisted counit. Another example is the set of conjugacy classes of a group. We also classify Frobenius objects in $\mathbf{\text{Rel}}$ with two or three elements, and we compute the associated surface invariants using the partition functions of the corresponding TQFTs.