Is it permissible for the questioner's father to repay the amount disbursed to him by mistake in monthly installments while retaining half of the remaining amount, knowing that his father is elderly and has limited income?

The duty was to notify the administration upon receiving the first salary, as a retired employee does not receive their full salary. Therefore, they must return the remaining amount and arrange with the administration to pay back what was disbursed in installments. Let a person know that their sustenance will undoubtedly come to them, and what Allah has decreed will be. So, one should not rush to acquire sustenance through unlawful means, for "And whoever fears Allah – He will make for him a way out And will provide for him from where he does not expect."