These fusion categories are all weakly integral, each with an FPdim less than 84, and therefore, they are all weakly group-theoretical by [this paper](https://doi.org/10.1016/j.aim.2010.06.009). Consequently, they can all be described using models that are, to varying degrees, derived from finite group theory.  

The first fusion ring is recognized as  N°3 of rank 6 in [this paper](https://doi.org/10.1007/s11005-022-01542-1), and should have a model from zesting.  

For the second one, see the last sentence of [this comment](https://mathoverflow.net/questions/369024/existence-of-twisted-metaplectic-categories/369169#comment933181_369169) by Eric Rowell (with $N=3$):   

> The other fusion rules could potentially be obtained as a
> Z2-equivariantization of the near-group categories of type Z/N +(N-1)
> in the Evans-Gannon notation. Just a guess, but the numerology seems
> to work out.

About the two last ones, what about zestings of $ch(Q_{16})$ and $ch(SD_{16})$?