  This paper mainly studies the ResLieDer pair in characteristic 2, that is, a
restricted Lie algebra with a restricted derivation. We define the restricted
representation of a ResLieDer pair and the corresponding cohomology complex. We
show that a ResLieDer pair is rigid if the second cohomology group is trivial
and a deformation of order $n$ is extensible if and only if its obstruction
class is trivial. Moreover, we prove that the central extensions of a ResLieDer
pair are classified by the second cohomology group. Finally, we show that a
pair of restricted derivations is extensible if and only if its obstruction
class is trivial.
