Is it permissible in Sharia to enter a car auction with a refundable security deposit, and if I win the auction, I pay an additional amount to the administration according to a declared schedule, in addition to the price of the car?

Auction sales are permissible, and there is no harm in them. There is also no harm in requiring participants to pay an upfront sum to enter the auction, provided that it is returned to those who do not win and is counted as part of the price for the winner. This sum serves as an assurance of the participant's seriousness. There is no harm in paying a fee for the terms and conditions booklet, if one exists, provided that the fee equals its actual cost. Furthermore, there is no harm in the administration taking a sum from the buyer, the seller, or both, in addition to the car's price, as this is a fee for brokerage and mediation services.