I would recommend you to use metadata API tools to deploy the permissions as the permissions will be reflected if the payload includes the corresponding components in it.
Please refer the below thread for the sample package.xml. https://salesforce.stackexchange.com/questions/2062/what-is-the-best-way-to-deploy-profiles-in-salesforce-especially-system-admin/19068
If are interested in using any other third party tool then you can consider this below blog which makes life easier.
https://www.salesforceben.com/an-easier-way-to-deploy-profiles-and-permissions-sets/Please mark it as best answer if it helps you to fix the issue.
Greetings!
I would recommend you to use metadata API tools to deploy the permissions as the permissions will be reflected if the payload includes the corresponding components in it.
Please refer the below thread for the sample package.xml.
https://salesforce.stackexchange.com/questions/2062/what-is-the-best-way-to-deploy-profiles-in-salesforce-especially-system-admin/19068
If are interested in using any other third party tool then you can consider this below blog which makes life easier.
https://www.salesforceben.com/an-easier-way-to-deploy-profiles-and-permissions-sets/Please mark it as best answer if it helps you to fix the issue.
Thank you!
Regards,
Shirisha Pathuri