  We consider the restricted Jordan plane in characteristic $2$, a
finite-dimensional Nichols algebra quotient of the Jordan plane that was
introduced by Cibils, Lauve and Witherspoon. We extend results from
\texttt{arXiv:2002.02514} on the analogous object in odd characteristic. We
show that the Drinfeld double of the restricted Jordan plane fits into an exact
sequence of Hopf algebras whose kernel is a normal local commutative Hopf
subalgebra and the cokernel is the restricted enveloping algebra of a
restricted Lie algebra $\mathfrak m$ of dimension 5. We show that $\mathfrak
u(\mathfrak m)$ is tame and compute explicitly the indecomposable modules. An
infinite-dimensional Hopf algebra covering the Drinfeld double of the
restricted Jordan plane is introduced. Various quantum Frobenius maps are
described.
