often email clients can download your inbox but can't easily import your sent emails.
you have to download a monolithic file (containing all the stuff in the selected folder) from your webmail server and to import it into your email client.

right-click on your sent email folder in your webmail window, then select export : the file will generally be saved without any extension. rename it adding the .mbox extension (eg. yoursavedsentemailname.mbox ).

simply download and install the mozilla thunderbird's add-on mbox import , then select the sent email folder, right-click on its label and select import/export - import mbox file , select the file previously downloaded from your webmail.

wait a minute and the add-on creates a new subfolder, named yoursavedsentemailname.mbox (number of messages imported) with all yuor stuff inside, with attachments!

this is tested on @aruba webmails.
this works with minor adjustments in every other email client (microsoft outlook, apple mail, lotus notes).

nota in italiano: come importare la cartella posta inviata di aruba o di qualsiasi client webmail in posta inviata di mozilla thunderbird.