Unoriented 2-dimensional TQFTs and the category $\operatorname{Rep}(S_t\wr \mathbb Z_2)$

We construct a family of unoriented 2-dimensional cobordism theories parametrized by certain triples of sequences. We also prove that some specializations of these sequences yield equivalences with an exterior product of Deligne categories. In particular, one of these specializations recovers the generalized Deligne category $\operatorname{Rep}(S_t\wr \mathbb Z_2)$.