I am trying to set up the out of office assistance as I will be out the office next week, however, the out of office message is only being sent when an e-mail is sent internally and not through an external e-mail. Could anybody assist me please?
To tell you what, specifically, to change will depend on knowing your version of Exchange.
Here's the procedure for Exchange 2000 to enable Out-of-Office messages to be sent to the Internet: http://support.microsoft.com/kb/262352
Here's the procedure for Exchange 2003 to enable Out-of-Office messages to be sent to the Internet: http://support.microsoft.com/kb/821899
For Exchange 2007, Out-of-Office functionality has changed dramatically and is enabled to the Internet by default.
More details of the email system you're using and possibly what you've tried already are customary at this point.
However, if you're using Exchange, try this (Exchange 2003, but its similar for all versions)
In the Exchange system manager expand the "Global Settings" node. Click on "Internet Message Formats" the right hand screen will change to display a "Default" entry. Right click on this and select properties - from the box that appears choose the "Advanced" tab - at the bottom you will see a tick box for "Allow out of office responses"- tick it, click apply. Give replication a chance to take place. Things should now be working.
Remember, when testing, OOO only allows one response to a mailbox per day, so don't keep trying with teh same external email address over and over, try testing with a few different ones.