Let $L$ be a perfect Lie algebra. Then it is well-known that $L$ has a universal central extension $\hat{L}$ and every derivation of $L$ can be lifted to a derivation of $\hat{L}$. (See e.g. Section 2 of https://mysite.science.uottawa.ca/neher/Papers/univ.pdf.)

Now suppose that $(\mathfrak{L},[p])$ is a restricted Lie algebra over a field of characteristic $p>0$. I remember that a derivation $D$ of $L$ is said to be restricted if $D(A^{[p]})=(\mathrm{ad} x)^{p-1}D(a)$ for every $a\in \mathfrak{L}$. If $\mathfrak{L}$ is perfect than $\mathfrak{L}$ admits a restricted universal central extension $\hat{\mathfrak{L}}$. (See e.g. http://rmi.tsu.ge/proceedings/volumes/pdf/v121-5.pdf.)

Is it true that every restricted derivation of $\mathfrak{L}$ can be lifted to a restricted derivation of $\hat{\mathfrak{L}}$?