It is not exactly what you are looking for, but it might be useful.
You can find an interesting construction of the triple covering for the alternating group $\mathbb{A}_6$ in Chapter 2, Section 7.3 of Wilson's book:
Wilson, Robert A. The finite simple groups. Graduate Texts in Mathematics, 251. Springer-Verlag London, Ltd., London, 2009. xvi+298 pp. ISBN: 978-1-84800-987-5 MR2562037 (2011e:20018)
Here you have the link (Google books).