I don't know whether Khandalyce was born in the first or second half of 2006, but if the first, she would have been due to start compulsory education in 2011. Possibly Centrelink sent out forms asking for school enrolment or other details because of Khandalyce's age. According to the Human Services website, the Schoolkids bonus is paid automatically to eligible families in January but you must first supply your child's enrolment details.
"From 1 January 2013, the schoolkids bonus replaces the Education Tax Refund that was previously paid through the tax system. Schoolkids bonus is paid in two instalments, in January and July each year, to assist eligible families and students with education expenses."
https://www.comlaw.gov.au/Details/F2012L02562/Explanatory%20Statement/Text
I gather that this wasn't applicable in 2010. From what I can gather, they used to get something called Education Tax Refund, and from what I understand, the child had to be in either primary or secondary school to be eligible.
file:///Users/imac/Downloads/IND00238018n726210610.pdf
I don't really understand it to be honest, it seems to say it's for people who are not required to lodge a tax return (which has me confused, for a start) but who want to claim a tax refund.
I don't think you would need to fill out any forms sent from Centrelink though.
I'm still trying to fathom why an imposter would have to front Centrelink in December 2010, but I haven't been able to find anything. As Khandalyce would have been around 4 at that time, Karlie wouldn't have been required to attend any job seeking (as far as I can determine), so unless there was a change of circumstances (like claiming disability), I can't think of any other reason they would need to attend, unless someone in Centrelink was suspicious and arranged a compulsory interview to alleviate their suspicions/concerns.
Anybody else come up with anything else, or got any ideas? I'm stumped.