Your service would fall under Sec 13(3)(a) of the IGST Act, which deals with Place of supply of services where location of supplier or location of recipient is outside India.
The relevant extract is
The place of supply of the following services shall be the location where the services are actually performed, namely:
(a) services supplied in respect of goods which are required to be made physically available by the recipient of services to the supplier of services, or to a person acting on behalf of the supplier of services in order to provide the services:
Hence, it is abundantly clear that in your case, the said service on the goods will be dealt with according to where they were repaired, and the same being in India would be subject to GST.
In relation to your second part of the query, the key component in order to avail ITC is to be registered under the GST Act, which I am assuming is not in the case of your overseas customer. Therefore, in this scenario, they cannot claim the ITC on the GST charged by you and the same would be a cost to either your overseas customer or to you depending upon the agreement executed.
Please advise in case of any other clarification.
Thanks & Regards,
CA Aditya Dhanuka.