  We provide an infinite family of counterexamples to the conjecture of
Zassenhaus on the solvability of the outer derivation algebra of a simple
modular Lie algebra. In fact, we show that the simple modular Lie algebras
$H(2;(1,n))^{(2)}$ of dimension $3^{n+1}-2$ in characteristic $p=3$ do not have
a solvable outer derivation algebra for all $n\ge 1$. For $n=1$ this recovers
the known counterexample of $\mathfrak{psl}_3(F)$. We show that the outer
derivation algebra of $H(2;(1,n))^{(2)}$ is isomorphic to
$(\mathfrak{sl}_2(F)\ltimes V(2))\oplus F^{n-1}$, where $V(2)$ is the natural
representation of $\mathfrak{sl}_2(F)$. We also study other known simple Lie
algebras in characteristic three, but they do not yield a new counterexample.
