  We investigate the theory of extending structures by the unified product for
perm algebras, and the factorization problem as well as the classifying
complements problem in the setting of perm algebras. For a special extending
structure, non-abelian extension, we study the inducibility of a pair of
automorphisms associated to a non-abelian extension of perm algebras, and give
the fundamental sequence of Wells in the context of perm algebras. For a
special extending structure, bicrossed product, we introduce the concept of
perm bialgebras, equivalently characterized by Manin triples of perm algebras
and certain matched pairs of perm algebras. We introduce and study coboundary
perm bialgebras, and our study leads to the ''$\mathcal{S}$-equation" in perm
algebras, which is an analogue of the classical Yang-Baxter equation. A
symmetric solution of $\mathcal{S}$-equation gives a perm bialgebra.
