  We explore the outcome of mass transfer via Roche lobe overflow (RLOF) of
$M_{\rm He}\lesssim0.51 M_\odot$ pure helium burning stars in close binaries
with white dwarfs (WDs). The evolution is driven by the loss of angular
momentum through gravitational wave radiation (GWR), and both stars are modeled
using Modules for Experiments in Stellar Astrophysics (MESA). The donors have
masses of $M_{\rm He}=0.35, 0.4, \&\ 0.51M_\odot$ and accrete onto WDs of mass
$M_{\rm WD}$ from $0.6M_\odot$ to $1.26M_\odot$. The initial orbital periods
($P_{\rm{orb}}$) span 20 to 80 minutes. For all cases, the accretion rate onto
the WD is below the stable helium burning range, leading to accumulation of
helium followed by unstable ignition. The mass of the convective core in the
donors is small enough so that the WD accretes enough helium-rich matter to
undergo a thermonuclear runaway in the helium shell before any carbon-oxygen
enriched matter is transferred. The mass of the accumulated helium shell
depends on $M_{\rm WD}$ and the accretion rate. We show that for $M_{\rm
He}\gtrsim0.4 M_\odot$ and $M_{\rm WD}\gtrsim0.8 M_\odot$, the first flash is
likely vigorous enough to trigger a detonation in the helium layer. These
thermonuclear runaways may be observed as either faint and fast .Ia SNe, or, if
the carbon in the core is also detonated, Type Ia SNe. Those that survive the
first flash and eject mass will have a temporary increase in orbital
separation, but GWR drives the donor back into contact, resuming mass transfer
and triggering several subsequent weaker flashes.
