I am currently working for a Federal Government department (on contract - wouldn't take a permant PS job!) in Canberra - I won't say which one but they do use Qantas Travel Services as their travel booking company. I don't work in the travel section but we share a printer and yesterday there was an unclaimed printout with numerous flight bookings on it. There were around a dozen bookings for a total of around 30 flights and every single flight was with QF - not a single Virgin or Rex sector.
Just out of interest I did a comparison of the fares paid on the first three bookings with what I could obtain on the internet for flights departing at around the same time with Qantas, Virgin and Rex (none of the sectors are flown by Jetstar).
Booking 1 - Melbourne/Canberra return - Ticketed price = $701
Cheapest - $409 using
VB to Canberra and QF return (no
VB flight at the required return time)
Booking 2 - Sydney/Canberra return - Ticketed price = $380
Cheapest - $199 using Rex to Canberra, QF return (QF $5 cheaper than Rex on return journey but $50 higher on other leg)
Booking 3 - Brisbane/Sydney/Canberra/Brisbane - Ticketed price = $1075
Cheapest = $365 using
VB for BNE/SYD and CBR/BNE, Rex on SYD/CBR
Total Ticket Value = $2156
Using Cheapest Fares = $973
So using the true "cheapest available" fares would save over 50% of the cost for this sample (and I have no doubt the overall travel budget).