All groups that be constructed from symmetric groups via cross products and wreath products have this property. See Section 3 of my paper "Mass formulas for local Galois representations to wreath products and cross products" http://arxiv.org/pdf/0804.4679v1.pdf

So, for example, $((S_7 \wr S_4) × S_3) \wr S_8$ has a rational character table. In fact, taking cross products and wreathing with $S_n$ preserves the property you are asking about (see above reference).

This includes several of the examples given: $(\mathbb Z/2\mathbb Z)^n,$ hyperoctahedral groups, and Sylow 2-subgroups of $S_n$. I am not sure if the index 2 subgroups of hyperoctahedral groups can be constructed from symmetric groups via cross products and wreath products.

*familyof groups with above property. In such groups, every conjugacy class is rational conjugacy class (see mathoverflow.net/questions/10635/…) $\endgroup$