Same Hall-Senior genus not implies character table-equivalent

Statement
It is possible to have two finite groups $$G_1$$ and $$G_2$$ such that:


 * 1) $$G_1$$ and $$G_2$$ are groups having the same Hall-Senior genus.
 * 2) $$G_1$$ and $$G_2$$ are not character table-equivalent groups, in fact, we can arrange our example so that the field generated by character values is different for $$G_1$$ and $$G_2$$.

Proof
The smallest example occurs with the three maximal class groups of order 16: dihedral group:D16, semidihedral group:SD16, and generalized quaternion group:Q16. It turns out that $$D_{16}$$ and $$Q_{16}$$ are character table-equivalent, but $$SD_{16}$$ is not character table-equivalent to either. The field generated by character values for the three groups are described below: