  We study the instabilities and related anti-evaporation of the extremal
Reissner-Nordstr\"om (RN) black hole in $F(R)$ gravity. It is remarkable that
the effective electric charge can be generated for some solutions of $F(R)$
gravity without electromagnetic field. The anti-evaporation effect occurs but
it emerges only in the strong coupling limit of the effective gravitational
coupling. The instabilities of RN black hole are also investigated when the
electromagnetic sector is added to the action of $F(R)$ gravity. We show the
anti-evaporation occurs in the Maxwell-$F(R)$ gravity with the arbitrary
gravitational coupling constant although it does not occur in the
Maxwell-Einstein gravity. Furthermore, general spherically-symmetric solution
of $F(R)$ gravity in the Einstein frame is obtained.
