  Horizontal branch (HB) morphology is a complex multiple-parameter problem.
Besides the metallicity, two other leading parameters are the mass loss rate
(MLR) and the initial He abundance of the HB progenitors. Using the STAREV
stellar evolution code, we produce a wide array of Extreme Horizontal Branch
(EHB) stars and also examine their post-HB evolution. EHB stars are produced in
our calculations by the so called `delayed (late) hot core flash' scenario. The
MLR is increased on the red giant branch (RGB) to the extent that, prior to
reaching core flash conditions, only a very thin H-rich envelope remains and
helium ignition takes place at hotter positions on the HRD. We perform an
extensive, self-consistent parameter study, covering populations I and II
(Z=0.0001-0.03), for both normal initial helium abundances and He-enriched
models (up to Y=0.40). For completeness of the study and in order to point to
complete trends, we chose NOT to cut out several combinations (or results of)
that may extend beyond realistic limits. We present results and complete
evolutionary tracks for the covered parameter space, showing in particular
that: a) Increased He abundance ON ITS OWN -- without having a
significant-enough MLR on the RGB -- DOES NOT lead to the production of EHB
stars; however, b) The bluest (hottest) HB positions do result from the
COMBINED effect of He-enhancement and increased MLR; c) The general trend is
that the effective temperature on the HB increases with decreasing metallicity,
but there is an indication for a halt, or even a reversal of this trend, as Z
further decreases below $10^{-3}$; d) EHB stars can serve as major contributors
to the UV flux emanating from their host system. Thus, the present
comprehensive study both complements and lends support to previous, more
restricted studies of the HB phase, and adds results for unexplored regions of
the parameter space.
