  We present a sample of 2865 emission line galaxies with strong nebular He II
4686 emissions in Sloan Digital Sky Survey Data Release 7 and use this sample
to investigate the origin of this line in star-forming galaxies. We show that
star-forming galaxies and galaxies dominated by an active galactic nucleus form
clearly separated branches in the He II 4686/H{\beta} versus [N II]
6584/H{\alpha} diagnostic diagram and derive an empirical classification scheme
which separates the two classes. We also present an analysis of the physical
properties of 189 star forming galaxies with strong He II 4686 emissions. These
star-forming galaxies provide constraints on the hard ionizing continuum of
massive stars. To make a quantitative comparison with observation we use
photoionization models and examine how different stellar population models
affect the predicted He II 4686 emission. We confirm previous findings that the
models can predict He II 4686 emission only for instantaneous bursts of 20%
solar metallicity or higher, and only for ages of ~ 4 - 5 Myr, the period when
the extreme-ultraviolet continuum is dominated by emission from Wolf-Rayet
stars. We find however that 83 of the star-forming galaxies (40%) in our sample
do not have Wolf-Rayet features in their spectra despite showing strong nebular
He II 4686 emission. We discuss possible reasons for this and possible
mechanisms for the He II 4686 emission in these galaxies.
