The feature of storing the “sent” message into your “Sent” folder is something that is specific to the mail client you are using.
This means that you should check all available options of the application from your phone and, in case you have no such option, opt to send a message to the app developer or to change the mail client you are using.
My old @Outlook.com address, my Work email and my Axigen email.
I tested sending emails from my @Outlook- and work email to my Axigen address.
Both of those appeared in the sent folder on my phone, and the sent folder on the webmail.
When I replied to the emails, from my phone, using the Axigen profile, the emails are not stored in the sent folder on the Axigen Webmail. (They were actually stored on the phone though)
I can not see any setting anywhere that has any setting for this.
The Axigen profile one is the only one using IMAP though