  In this paper, we study non-abelian extensions of strict Lie 2-algebras via
the cohomology theory. A non-abelian extension of a strict Lie 2-algebra $\g$
by $\frkh$ gives rise to a strict homomorphism from $\g$ to $\SOut(\frkh)$.
Conversely, we prove that the obstruction of existence of non-abelian
extensions of strict Lie 2-algebras associated to a strict Lie 2-algebra
homomorphism from $\g$ to $\SOut(\frkh)$ is given by an element in the third
cohomology group. We further prove that the isomorphism classes of non-abelian
extensions of strict Lie 2-algebras are classified by the second cohomology
group.
