That is rather a long winded way of saying that your original "Government Departments never back down and apologise, even when they've done something outrageous." was indeed just your own personal opinion based on nothing.

Your requested example (Oddly enough from the Department for Works and Pensions):

In August 2015 the department admitted using fictional stories from made-up claimants on leaflets advertising the positive impact of benefit sanctions, following a Freedom of Information request from Welfare Weekly, claiming that they were for "illustrative purposes only" and that it was "quite wrong" to pass these off as genuine quotes.

Sources verifying the above:
- Dial2Donate. 26 August 2015.
- The Guardian Kevin Rawlinson; Frances Perraudin 18 August 2015.
- The Guardian Kevin Rawlinson 21 August 2015.
- The Guardian - The Minister for Work and Pensions, Iain Duncan-Smith, admits a leaflet about benefits containing fake quotes from fictitious claimants was 'wrong' Andrew Sparrow (24 August 2015).