The above is certainly possible, however some suppliers do charge a fee to take a vehicle across the border into Switzerland.
When searching for car hire from the Swiss side of Geneva airport, prices are generally higher than the French side and for the sake of a short walk through passport control, you can generally save yourself a decent amount of holiday spends.
To make sure you are booking the package that best suits your needs, we can look at both sides of the airport for you. Please note however, if you plan to book online yourself. Some suppliers charge an additional fee to do this. To make it easier and to avoid booking with the wrong supplier, call us on (0044) 113 2899 281 or email us at sales@thealternativeagency.co.uk.