  In (arXiv:1004.4127) the concept of down-link from a (K_v, G)-design B to a
(K_n, G')-design B' has been introduced. In the present paper the spectrum
problems for G'= P4 are studied. General results on the existence of
path-decompositions and embeddings between path- decompositions playing a
fundamental role for the construction of down-links are also presented
