Is it permissible to perform external work tasks using one's private car and then take the taxi fare allocated for that purpose?

If the company's system provides a fixed allowance for work-related travel, then it is permissible to use one's private car and take the allowance. However, if the system requires payment of actual costs, then it is not permissible to use one's private car and charge the company a fare unless with its permission. The employee must regulate this so as not to wrong the employer or charge them more than what a taxi would cost.