  This study builds on previous works, producing the most extensive prediction
of the properties of such a hypothetical population to date, taking into
account both Chandrasekhar and non-Chandrasekhar mass events. These results are
then used to define criteria for membership of this population and characterise
putative subpopulations
  This study contains 6x10^6 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.
  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. We find that current observations support a
Galactic SN rate on the order of ~3x10^-7/yr to ~2x10^-6/yr, 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. The number of
currently observed population members suggests that the He-donor scenariois not
a dominant contributor to the number of observed SNe Ia. However, we find that,
even at the low event rate suggested, 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. - abridged -
