Hi @Berni Horvath
Thank you for posting your question in the Microsoft Q&A forum.
When you try sending email and it fails, do you receive any bounce back email?
Usually, it is an NDR (non-delivery report), it contains an error code that explains the cause.
If you receive it, please provide it here so we can check it for you.
Additionally, please ask your IT admin to perform a message trace.
Message trace follows email messages as they travel through your Microsoft 365 organization. You can determine if the service received, rejected, deferred, or delivered a message.
Please follow the steps here: Message trace in the new EAC in Exchange Online | Microsoft Learn
Ask your IT if they have set any transport rules to restrict sending emails to external recipients.
Please provide more information so I can assist you more effectively.
Thanks for your cooperation.