  Stability of a half-quantum vortex (HQV) in superfluid $^3$He has been
discussed recently by Kawakami, Tsutsumi and Machida in Phys. Rev. B {\bf 79},
092506 (2009). We further extend this work here and consider the A$_2$ phase of
superfluid $^3$He confined in thin slab geometry and analyze the HQV realized
in this setting. Solutions of HQV and singly quantized singular vortex are
evaluated numerically by solving the Ginzburg-Landau (GL) equation and
respective first critical angular velocities are obtained by employing these
solutions. We show that the HQV in the A$_2$ phase is stable near the boundary
between the A$_2$ and A$_1$ phases. It is found that temperature and magnetic
field must be fixed first in the stable region and subsequently the angular
velocity of the system should be increased from zero to a sufficiently large
value to create a HQV with sufficiently large probability. A HQV does not form
if the system starts with a fixed angular velocity and subsequently the
temperature is lowered down to the A$_2$ phase. It is estimated that the
external magnetic field with strength on the order of 1 T is required to have a
sufficiently large domain in the temperature-magnetic field phase diagram to
have a stable HQV.
