We introduced, for any set S, the concept of K-family between two Hilbert C∗-modules over two C∗-algebras, for a given completely positive definite (CPD-) kernel K over S between those C∗-algebras and obtained a factorization theorem for such K-families. If K is a CPD-kernel and E is a full Hilbert C∗-module, then any K-family which is covariant with respect to a dynamical system (G, η, E) on E, extends to a K ̃-family on the crossed product E×ηG, where K ̃ is a CPD-kernel. Several characterisations  of k-families, under the assumption the E is full, were obtained and covariant versions of these results were also given . One of these characterisation says that sunch k-families extend as CPD-kernels between associated (extended) linking algebras, whose (2,2)-corner is a homomorphism and  vice versa. This leads to new insights in the dilation theory odf CPD-kernels in relation to k-families. Given an aotomorphism α on a C*-algebra, we obtained a Kolmogorov decomposition of α-completely positive definite kernels and investigated a notion of distance between α-CPD-kernels called the Bures distance α-CPD-kernels. We also defined transition probability for these kernels and found a characterisation of the transition probability.