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Since this debt is not the responsibility of your children, it does not belong on Schedule H, as your children are not co-debtors for this debt.
Additionally, this creditor cannot try to collect the debt from your children, even though the debt is listed on one credit report - where there should be a notation of "authorized user".
Two of the credit bureaus will remove the debt if your children call the credit bureau to point out the fact that he or she is only an authorized user, and this debt should therefore not be on their credit report. One of the credit bureaus (I believe it is Transunion) will only remove the debt if the creditor removes them as an authorized user. For this credit bureau, then, you would have to call the creditor and ask them to remove your children as authorized users - and then the debt will be removed from their credit report.