Why do I sometimes lose my two gmail accounts in Apple Mail when the system goes to sleep and then is awakened?

I have my iMac configured to go to sleep when not used after a certain period of time. Every once in a while when I wake it, my two gmail accounts are not active. It is getting harder and harder to get them back on line. I get a message that I need to login to my Google account. I've tied logging off and then logging in to both of my Google accounts and still can't get my two gmail accounts to go on line. My Apple mail account in just fine.

What can I do to have my gmail accounts not go off line when my system goes to sleep (doesn't happen every time).

I thought I was running the latest MacOS Big Sur 11.6. I see that an update to Big Sur 11.6.1 is now available I was not prompted to do an update. I'll try doing the update.

The system will go to sleep after sitting idle for a few hours. It nearly always comes awake when I hit a key on the keyboard. Once or twice recently I've had to restart the computer to get it to wake up, doesn't happen often.

The following settings are checked in Energy Saver:

Turn display off after 15 min.

Prevent computer from sleeping automatically when the display is off.

Put hard disks to sleep when possible

Wake for network access.

The other two boxes are unchecked.

Time Machine continues to make hourly backups while the system sleeps.

Right now I cannot get either of my gmail accounts to go on line. I've tried all the options in the Mail menus. I've logged Google off and logged back on, using both of my Google accounts, one at a time. It is now 12:25 pm and the last mail I received was last night at 5:40 pm. The iMac sat idle until about 30 minutes ago. I've tried restarting MacOS. Still can't get my google mail accounts to go on line.

Any suggestions???

Thanks

veehbj