It's actually the style of email you have. As long as you're on a POP style email server, it will continue to work that way, I would recommend you suggest they start using an IMAP configuration. I'll try to give a somewhat coherent breakdown below between the two....
POP EMAIL
Receive message ---> goes to which ever device sees it on the server first.
Example: If you have your phone on, and setup with your email, but your computer is off or Outlook is closed-- you will see the message on your iPhone, but not your Outlook email client (or vice versa). The message is only populated on one receiving device so you may only see the email in one inbox (say on your phone) and not on your computer.
IMAP EMAIL
Receive message --->> goes to both devices.
Example: If you have the iPhone and Outlook setup to receive, both will see the message at the same time. So you can view the message on either device. When you delete the message it should delete from both devices.
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Outlook also has a setting to remove the email from the server upon receiving it. Make sure this is turned off for IMAP. Other wise it will still work sorta like a POP email account. Outlook is merely a tool to fetch mail from an email server, not to be confused with the actual email server itself.
Anyone else feel free to correct or add on to my post. I only know this stuff because I worked tech support for a huge web/domain registrar for some time. I'd say roughly 60% of my tech support calls were email related. My personal opinion is that Outlook is overrated as an email client, too many people rely on it with false hopes of greatness. I despise/loathe it. I would always recommend an IMAP account if you only are looking to do email. Outlook is only descent in conjunction with an actual Exchange server setup, that's when the calendar features and other add-ons make Outlook worth using.