[Grhm]

There may be a better solution, but I've tried this out, and it works...

In the desktop version of webmail
• Mail > Settings > Users & Aliases
• Enter password & click 'Unlock'
• Click 'New alias'
• Enter a new, hard-to-guess alias,
e.g. massfwd.958336964 @ yrdomain.co
• Remove the 'Deliver to' address(es)
• Tick the 'Also send to all contacts in' checkbox
• Select the relevant mass-forwarding contact group
• Click 'Save'
• Settings > Contacts > New contact
• Enter a simple name e.g. MassFwd
• Enter your hard-to-guess alias as the address
• Click 'Save'
• Wait for 15 minutes for the alias to propagate

Now when you have a mail you want sent to your mass-forwarding contact group, leave the To and Cc fields blank and just type 'MassFwd' into the Bcc field to call up that hard-to-guess alias.
Do not add the name of the contact group nor add any other addresses.
Recipients will see the address you sent it from, but not the secret alias nor any other addresses.