Context. Thermonuclear supernovae (SNe), a subset of which are the highly important SNe of Type Ia and Iax, are relatively poorly understood phenomena. One of the more promising scenarios leading up to the creation of a thermonuclear SN involves accretion of helium-rich material from a binary companion. Following the SN, the binary companion is then ejected from the location of the progenitor binary at velocities possibly large enough to unbind it from the gravitational potential of the Galaxy. Ejected companion stars should form a detectable population, if their production mechanism is not exceedingly rare. Aims. This study builds on previous works, producing the most extensive prediction of the properties of such a hypothetical population to date, taking both Chandrasekhar and non-Chandrasekhar mass events into account. These results are then used to define criteria for membership of this population and characterise putative subpopulations. Methods. This study contains 6 x 10(6) individual ejection trajectories out of the Galactic plane calculated with the stellar kinematics framework SHyRT, which are analysed with regard to their bulk observational properties. These are then put into context with the only previously identified population member US 708 and applied to a number of other possible candidate objects. Results. We find that two additional previously observed objects possess properties to warrant a designation as candidate objects. Characterisation of these object with respect to the predicted population finds all of them to be extreme in at least one astrometric observable. Higher mass ( >0 :7 M-circle dot) objects should be over-represented in the observationally accessible volume, with the ratio of bound to unbound objects being an accessible observable for the determination of the dominant terminal accretor mass. We find that current observations of runaway candidates within 10 kpc support a Galactic SN rate of the order of similar to 3 x 10(-7) yr(-1) to similar to 2 x 10(-6) yr(-1), three orders of magnitude below the inferred Galactic SN Ia rate and two orders of magnitude below the formation rate of predicted He-donor progenitors. Conclusions. The number of currently observed population members suggests that the He-donor scenario, as suspected before, is not a dominant contributor to the number of observed SNe Ia. However, even at the low event rate suggested, we find that the majority of possibly detectable population members is still undetected. The extreme nature of current population members suggests that a still larger number of objects has simply evaded detection up to this point, hinting at a higher contribution than is currently supported by observation.