It isn't clear if you think "receiving a refund" is what changes the taxable status. Let's think of it as "entitled to a refund" and then they get to decide what to do with the money: "receive it" = get it sent to them; or, "roll it" = let the agency treat it as applied to the next cycle for what they would have paid from their own bank account.

Nothing here changes if it is taxable to them. That determination is made by using the lookback to the year the refund is from. If they itemized in that year, they deducted too much (because they clearly paid too much) and this refund is treated as the opposite of deduction = income for the year received.