Hello.
My issue is this.

Every time I put a player in a group for a limited time with the command:
pex user {name} group add fly "" 3600

This will put the player into the fly group for 1 hour. But, if the server restarts during this time the person will be removed from this group!

We sometimes give out a full week of fly on our server and in this case we have to start it every restart.

Specs:
Spigot 1.8.7
Java 7
Pex Version 1.23.3

How can I fix this?

Timed permissions are the only ones stored in memory -- timed groups should be written to the file. My only guess is that a file save is not yet complete when you restart the server? -- Check the permissions file to see whether or not the group-fly-until option is set on the user.

I add myself to the group.

I go to the file to check!

It's there!

I'm actually still in the group after the restart.

The title is wrong.
What I actually meant was when I add a timed permission to a group it's removed after a restart.

Edited the title for you.

That's expected behavior -- timed permissions are in-memory only -- only timed groups are stored, so you may want to make a timed container group that will better meet your needs.