Hi ymdema,
There are ways to get a visa for your wife and or your family if you really want to stay and live full time in Japan, a few guys have done it over the years. The company (AJX) will not assist in anyway though, and you will have to sort that out by yourself. There are a couple of lawyers in Narita that are very good with setting this up and speak very good English that can help. Different options involve setting up Japanese companies (surprisingly easy to do) or student visas for your kids (requiring Adult supervisor visas) etc......International schooling can be expensive here though, so think long and hard about it if you have a number of children. Commuting to Italy is always another option.
Cheers.