  We obtain the $\epsilon<2$ new extremal ground states of a two-dimensional
(2D) charged black hole where $\epsilon$ is the dilaton coupling parameter for
the Maxwell term. The stability analysis is carried out for all these extremal
black holes. It is found that the shape of potentials to an on-coming tachyon
(as a spectator) take all barrier-well types. These provide the bound state
solutions, which imply that they are unstable. We conclude that the 2D,
$\epsilon<2$ extremal black holes should not be considered as a toy model for
the stable endpoint of the Hawking evaporation.
