Simple one here – a VIP relies on a Gmail account when the company is on Office365. To allow the VIP to send to a distribution list, I had to do the following:
1) Open Exchange Admin Center
2) Click on Recipients and then Groups
3) Select the distribution group
4) Click the edit button 
5) Click Delivery Management and select “Senders inside and outside of my organization”
6) Then just click Save! 
You can also use this area to block or allow certain senders to distribution lists by adding emails to the box below.