Solution 1:

This answer utilizes plugins, and is not applicable to vanilla servers.

I'm not as familiar with GroupManager, but it seems that the plugin does not offer a "temporary rank" feature.

Considering you can use plugins, use the timed permissions feature offered by PermissionsEX.

pex user <user> group add <group> [world] [time]

where time parameter is measured in seconds. To add the group to the user globally, use * in the world parameter.

Alternatively, there are many plugins dedicated to timed permissions and groups.

Note 1: PEX is unmaintained at the moment, but it works for most servers. A popular alternative is LuckPerms which has a migration feature.

Note 2: Nitro Server Boosts don't expire after 7 days. A server boost stays indefinitely until the booster chooses to boost another server.