Ask HN: How to “alias” an email with a different provider?

I have an email account, provided by a mobile/internet provider, which can only be used when connected to the said operator, either via mobile or from a fixed connection. Use = receive incoming emails or send them. The SMTP server only accepts requests when I am connected to the provider 4G or fixed internet connection.

I would like to keep using the account, but basically just forward all emails to a gmail account (easy) and send emails from the gmail account, but forwarded through my old email.

I can use a raspberry pi connected to the home network which is run by the said provider.

Any ideas?