  In recent years, averaging operators on Lie algebras (also called embedding
tensors in the physics literature) and associated tensor hierarchies form an
efficient tool for constructing supergravity and higher gauge theories. A Lie
algebra with an averaging operator is called an averaging Lie algebra. In the
present paper, we introduce $2$-term averaging $L_\infty$-algebras and give
characterizations of some particular classes of such homotopy algebras. Next,
we study non-abelian extensions of an averaging Lie algebra by another
averaging Lie algebra. We define the second non-abelian cohomology group to
classify the equivalence classes of such non-abelian extensions. Next, given a
non-abelian extension of averaging Lie algebras, we show that the obstruction
for a pair of averaging Lie algebra automorphisms to be inducible can be seen
as the image of a suitable Wells map. Finally, we discuss the Wells short exact
sequence in the above context.
