Are they citizen? if yes, they can study in the public school at the same district as your 親戚 live. Lucky that the Summer holiday is Dec and Jan in Australia. Mean your kid can study from June to August in regular class.
Unlike Hong Kong. This is LAW, the school must accept your kid even study 2 months. The "no seat" is not excuse.
I know some schools do welcome students visiting from overseas for a short period like this. At least this is true for my kid's primary school in Melbourne but I am not sure about Sydney. I guess it depends on each school's policy and resources. You can try to ask the schools near your relative's home. Also if your relative also has kids attending school, it might be easier to ask them to refer you to the same school.