  In this work the finite $\beta$-effects of an electron branch of the geodesic
acoustic mode (el-GAM) driven by electron temperature gradient (ETG) modes is
presented. The work is based on a fluid description of the ETG mode retaining
non-adiabatic ions and the dispersion relation for el-GAMs driven non-linearly
by ETG modes is derived. The ETG growth rate from the fluid model is compared
to the results found from gyrokinetic simulations with good agreement. A new
saturation mechanism for ETG turbulence through the interaction with el-GAMs is
found, resulting in a significantly enhanced ETG turbulence saturation level
compared to the mixing length estimate. It is shown that the el-GAM may be
stabilized by an increase in finite $\beta$ as well as by increasing
non-adiabaticity. The decreased GAM growth rates is due to the inclusion of the
Maxwell stress.
