Here is information from the Trenitalia website which has NOT been translated into English (it is available only on the Italian-language webpages):
NEW "FAMILIA ESTATE" Promotional Fare
DIFFERENT from (and more generous than) the normal "Familia" promotional fare, it is:
Available ONLY from July 1 through August 31
Available on ANY of the following train categories: AV-Fast, AV, ES-Fast, ES, ES-City (i.e. NOT IC, ICN, EXP, EC, ECN or "regionale" trains)
CAPACITY CONTROLLED
THREE to FIVE passengers travelling together (i.e. NOT two and NOT six)
At least one adult and one child under 12
Children under 12 travel FREE FREE FREE
Adults and teens (i.e. anybody over 12) get a 20% discount
NO ADVANCE PURCHASE REQUIRED (but capacity controlled)
NO RESERVATION CHANGES ALLOWED AFTER SCHEDULED TRAIN DEPARTURE
10 The "regular" "Familia" promotional fare CONTINUES to be available as a less generous ALTERNATIVE to the "Familia Estate" fare. The "Familia" fare ALSO covers IC, ICN and EXP trains).
- Don't confuse these two family promotional fares
NOTE: If NEITHER "familia" fare is available for your departure days and times (because both fares are capacity controlled), children under 12 nevertheless ALWAYS get a 50% discount, and toddlers under 4 nevertheless ALWAYS travel free. Remember this rule about children and toddlers!!!
When travelling as a family group of three to five, ALWAYS ask about these two family fares. If there are more than five in your travelling party, no problem: request the "familia" fare for five persons, and the "excess" family members get their own "regular fare" tickets.