Is the car that the father buys permissible if the costs of the extra day he spends after a business trip are at the expense of the association, or if he is not held accountable for a day he is absent from work, or if the delay in his return causes an increase in the price of the travel ticket? And must he inform the Board of Directors or anyone from the association about what he intends to do?

Charitable organizations act as agents for donors, so it is not permissible to spend their money on anything other than what they have specified, nor on personal interests for the employees. The car in the father's possession is not rendered unlawful, even if the father's action was prohibited. The father's informing the charity's administration of what he intends to do depends on the charity's system and regulations, based on the general Hadith (definition): "Muslims are bound by their conditions."