Call Forwarding through Office to iPhone

We can set up call forwarding from our office desk phone to our cell phone and it works fine for Android phone users -- they can see their contact name come through when the call is forwarded to their cell. But, for iPhone users, we only see the generic office phone number from where the call originally came through before forwarding -- we never know who is calling us. Has anyone had this experience and found a resolution or does anyone even have an idea of where to start researching a solution?

We have heard that it's because of the way iPhones save contacts and transfer them when you get a new phone, so there are too many and it gets clogged up, but we have cleaned our contact list thoroughly and it still just shows the office number, not the name of our contacts.