In this talk we discuss the conformal fractional powers of the horizontal Laplacian in groups of Heisenberg type. We present a new approach, based on the heat kernels of suitable extension operators, to the derivation of the fundamental solutions for these nonlocal operators and to the explicit construction of the Aubin-Talenti type functions. Frequent comparisons with the Euclidean case of the powers of the Laplacian will be made, focusing on similarities and main differences. The talk is based on a joint project with N. Garofalo.