Rational not implies rational-representation

Statement
It is possible to have a group (specifically, a finite group) that is a rational group (in the sense that all its character values are rational) but is not a rational-representation group (in the sense that not all its irreducible representations over the complex numbers can be realized over the rationals).

Such an example group must have at least one irreducible linear representation with Schur index greater than $$1$$.

Example of the quaternion group
The quaternion group of order eight is an example. It has a two-dimensional irreducible representation over the complex numbers with a rational character, but which cannot be realized over the rationals. The double of this representation can be realized over the rationals.