I can synch emails I post on iPhone, iPad and iMac. But when it comes to deleting, there is no synch and I have to delete form each device in turn.

Hi there ProfGold,

We'd like to help get those deleted emails to sync across all three devices, so let's start with ruling out some basics:

1. From your post, we see that you're running macOS 11.5, but the latest version is macOS 11.6. You can verify this by clicking on the Apple logo at the top left corner of the screen, then on ‘About This Mac’. Here's how to first Back up your Mac with Time Machine, and afterwards, here's How to update the software on your Mac.
2. If you have a compatible iPad and iPhone, the latest version is iPadOS/iOS 15.0.2. You can verify this within Settings > General > About. Here's a link that'll show how to first back up, then Update your iPhone, iPad, or iPod touch.
3. If you already have the aforementioned versions installed on each device, have you already tried any steps, like a restart as one example?
4. Has this worked in the past without any issues, and if yes, when did this first start?
5. Does it seem to be affecting multiple email accounts, or only one from which you delete emails?

Take care!