  The notion of Lie $H$-pseudoalgebra is a higher-dimensional analogue of Lie
conformal algebras. In this paper, we classify the equivalence classes of
non-abelian extensions of a Lie $H$-pseudoalgebra $L$ by another Lie
$H$-pseudoalgebra $M$ in terms of the non-abelian cohomology group $H^2_{nab}
(L, M)$. We also show that the group $H^2_{nab} (L, M)$ can be realized as the
Deligne groupoid of a suitable differential graded Lie algebra. Finally, we
consider the inducibility of a pair of Lie $H$-pseudoalgebra automorphisms in a
given non-abelian extension. We show that the corresponding obstruction can be
realized as the image of a suitable Wells map in the context.
